Search: a046523 -id:a046523
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1, 5, 2, 18, 2, 23, 2, 59, 7, 23, 2, 94, 2, 23, 16, 195, 2, 80, 2, 94, 16, 23, 2, 355, 7, 23, 29, 94, 2, 467, 2, 672, 16, 23, 16, 706, 2, 23, 16, 355, 2, 467, 2, 94, 67, 23, 2, 1331, 7, 80, 16, 94, 2, 302, 16, 355, 16, 23, 2, 1894, 2, 23, 67, 2422, 16, 467, 2, 94, 16, 467, 2, 2779, 2, 23, 67, 94, 16, 467, 2, 1331, 121, 23, 2, 1894, 16, 23, 16, 355, 2, 1832
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OFFSET
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1,2
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function
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FORMULA
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a(n) = (1/2)*(2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n)).
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PROG
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(PARI)
A001511(n) = (1+valuation(n, 2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A286161(n) = (2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286161.txt", n, " ", A286161(n)));
(Scheme) (define (A286161 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A046523 n)) 2) (- (A001511 n)) (- (* 3 (A046523 n))) 2)))
(Python)
from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a046523(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a046523(n)) # Indranil Ghosh, May 06 2017
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CROSSREFS
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Cf. A000027, A001511, A046523, A286160, A286162, A286163, A286164, A286251.
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, May 04 2017
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STATUS
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approved
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1, 2, 4, 2, 8, 6, 16, 2, 4, 12, 32, 6, 64, 24, 12, 2, 128, 6, 256, 12, 36, 48, 512, 6, 8, 96, 4, 24, 1024, 30, 2048, 2, 72, 192, 24, 6, 4096, 384, 144, 12, 8192, 60, 16384, 48, 12, 768, 32768, 6, 16, 12, 288, 96, 65536, 6, 72, 24, 576, 1536, 131072, 30, 262144, 3072, 36, 2, 216, 120, 524288, 192, 1152, 60, 1048576, 6, 2097152, 6144, 12, 384, 48, 240, 4194304, 12, 4
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1, 2, 5, 12, 14, 23, 27, 59, 42, 40, 65, 109, 90, 61, 86, 261, 152, 142, 189, 179, 148, 115, 275, 473, 273, 148, 318, 265, 434, 674, 495, 1097, 320, 226, 430, 1093, 702, 271, 430, 757, 860, 832, 945, 485, 619, 373, 1127, 1969, 1032, 485, 698, 619, 1430, 838, 1030, 1105, 856, 556, 1769, 2791, 1890, 625, 1117, 4497, 1426, 1196, 2277, 935, 1220
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OFFSET
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1,2
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function
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FORMULA
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a(n) = (1/2)*(2 + ((A000010(n)+A046523(n))^2) - A000010(n) - 3*A046523(n)).
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PROG
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(PARI)
A000010(n) = eulerphi(n);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A286160(n) = (2 + ((A000010(n)+A046523(n))^2) - A000010(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286160.txt", n, " ", A286160(n)));
(Scheme)
(define (A286160 n) (* (/ 1 2) (+ (expt (+ (A000010 n) (A046523 n)) 2) (- (A000010 n)) (- (* 3 (A046523 n))) 2)))
(Python)
from sympy import factorint, totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a046523(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1
def a(n): return T(totient(n), a046523(n)) # Indranil Ghosh, May 06 2017
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CROSSREFS
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Cf. A000010, A000027, A046523, A286161, A286162, A286163, A286164.
Cf. for example A061468 (one of the sequences this matches with).
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, May 04 2017
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STATUS
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approved
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A278223
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Least number with the same prime signature as the n-th odd number: a(n) = A046523(2n-1).
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+20
17
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1, 2, 2, 2, 4, 2, 2, 6, 2, 2, 6, 2, 4, 8, 2, 2, 6, 6, 2, 6, 2, 2, 12, 2, 4, 6, 2, 6, 6, 2, 2, 12, 6, 2, 6, 2, 2, 12, 6, 2, 16, 2, 6, 6, 2, 6, 6, 6, 2, 12, 2, 2, 30, 2, 2, 6, 2, 6, 12, 6, 4, 6, 8, 2, 6, 2, 6, 24, 2, 2, 6, 6, 6, 12, 2, 2, 12, 6, 2, 6, 6, 2, 30, 2, 4, 12, 2, 12, 6, 2, 2, 6, 6, 6, 24, 2, 2, 30, 2, 2, 6, 6, 6, 12, 6, 2, 6, 6, 6, 6, 6, 2, 36, 2, 2
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OFFSET
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1,2
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COMMENTS
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This sequence works as a filter for sequences related to the prime factorization of odd numbers by matching to any sequence that is obtained as f(2*n - 1), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ..."). The last line in Crossrefs section lists such sequences that were present in the database as of Nov 11 2016, although some of the matches might be spurious.
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences computed from exponents in factorization of n
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FORMULA
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a(n) = A046523(2n - 1).
a(n) = A046523(A064216(n)).
From Antti Karttunen, May 31 2017: (Start)
a(n) = A278222(A244153(n)).
a(n) = A278531(A245611(n)).
(End)
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PROG
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(Scheme)
(define (A278223 n) (A046523 (+ n n -1)))
(define (A278223 n) (A046523 (A064216 n)))
(Python)
from sympy import factorint
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a046523(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1
def a(n): return a046523(2*n - 1) # Indranil Ghosh, May 11 2017
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CROSSREFS
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Odd bisection of A046523.
Cf. A064216, A244153, A245611, A278222, A278224, A278531.
Sequences that partition or seem to partition N into same or coarser equivalence classes: A099774, A100007, A193773, A101871, A101875, A158280, A158315, A158647, A285716.
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, Nov 16 2016
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STATUS
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approved
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2, 5, 12, 14, 23, 42, 38, 44, 40, 61, 80, 117, 80, 84, 216, 152, 23, 148, 80, 148, 601, 142, 302, 375, 109, 142, 911, 183, 302, 1020, 530, 560, 61, 61, 142, 856, 467, 142, 412, 430, 467, 1741, 1832, 265, 2497, 412, 1178, 1323, 109, 265, 826, 265, 1832, 1735, 2932, 489, 412, 412, 2630, 2835, 1178, 672, 2787, 2144, 61, 625, 80, 148, 601, 850, 302, 2998, 467, 601
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OFFSET
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1,1
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function
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FORMULA
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a(n) = (1/2)*(2 + ((A046523(n)+A278222(n))^2) - A046523(n) - 3*A278222(n)).
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PROG
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(PARI)
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A286163(n) = (2 + ((A046523(n)+A278222(n))^2) - A046523(n) - 3*A278222(n))/2;
for(n=1, 10000, write("b286163.txt", n, " ", A286163(n)));
(Scheme) (define (A286163 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A278222 n)) 2) (- (A046523 n)) (- (* 3 (A278222 n))) 2)))
(Python)
from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a046523(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1
def a278222(n): return a046523(a005940(n + 1))
def a(n): return T(a046523(n), a278222(n)) # Indranil Ghosh, May 05 2017
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CROSSREFS
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Cf. A000027, A046523, A278222, A286160, A286161, A286162, A286164.
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, May 04 2017
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STATUS
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approved
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1, 2, 4, 2, 4, 8, 6, 2, 4, 12, 16, 8, 6, 12, 6, 2, 4, 12, 36, 12, 16, 32, 24, 8, 6, 30, 24, 12, 6, 12, 6, 2, 4, 12, 36, 12, 36, 72, 60, 12, 16, 48, 64, 32, 24, 72, 24, 8, 6, 30, 60, 30, 24, 48, 60, 12, 6, 30, 24, 12, 6, 12, 6, 2, 4, 12, 36, 12, 36, 72, 60, 12, 36, 180, 144, 72, 60, 180, 60, 12, 16, 48, 144, 48, 64, 128, 96, 32, 24, 120, 216, 72, 24, 72
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OFFSET
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0,2
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 0..65537
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FORMULA
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a(n) = A046523(A243353(n)).
a(n) = A278222(A003188(n)).
a(n) = A278220(1+A075157(n)).
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MATHEMATICA
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f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; g[n_] := If[n == 1, 1, Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]];
Table[g@ f[BitXor[n, Floor[n/2]], 1, 1], {n, 0, 93}] (* Michael De Vlieger, May 09 2017 *)
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PROG
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(Scheme) (define (A278219 n) (A046523 (A243353 n)))
(Python)
from sympy import prime, factorint
import math
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a046523(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1
def a003188(n): return n^int(n/2)
def a243353(n): return a005940(1 + a003188(n))
def a(n): return a046523(a243353(n)) # Indranil Ghosh, May 07 2017
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CROSSREFS
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Cf. A003188, A046523, A075157, A243353, A278220.
Other base-2 related filter sequences: A278217, A278222.
Sequences that (seem to) partition N into same or coarser equivalence classes are at least these: A005811, A136004, A033264, A037800, A069010, A087116, A090079 and many others like A105500, A106826, A166242, A246960, A277561, A037834, A225081 although these have not been fully checked yet.
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, Nov 16 2016
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STATUS
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approved
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A300226
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Filter sequence combining A046523(n) and A052126(n), the prime signature of n and n/(largest prime dividing n).
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+20
15
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1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 8, 9, 2, 10, 2, 7, 8, 4, 2, 11, 12, 4, 13, 7, 2, 14, 2, 15, 8, 4, 16, 17, 2, 4, 8, 11, 2, 14, 2, 7, 18, 4, 2, 19, 20, 21, 8, 7, 2, 22, 16, 11, 8, 4, 2, 23, 2, 4, 18, 24, 16, 14, 2, 7, 8, 25, 2, 26, 2, 4, 27, 7, 28, 14, 2, 19, 29, 4, 2, 23, 16, 4, 8, 11, 2, 30, 28, 7, 8, 4, 16, 31, 2, 32, 18, 33, 2, 14, 2, 11, 34
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OFFSET
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1,2
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COMMENTS
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Restricted growth sequence transform of P(A046523(n), A052126(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..65537
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EXAMPLE
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a(6) = a(10) (= 4) because both are nonsquare semiprimes (2*3 and 2*5), and when the largest prime factor is divided out, both yield the same quotient, 2.
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PROG
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(PARI)
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A052126(n) = if(1==n, n, my(f=factor(n)[, 1], gpf = f[#f]); n/gpf); \\ After code in A052126.
Aux300226(n) = (1/2)*(2 + ((A052126(n)+A046523(n))^2) - A052126(n) - 3*A046523(n));
write_to_bfile(1, rgs_transform(vector(65537, n, Aux300226(n))), "b300226.txt");
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CROSSREFS
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Cf. A046523, A052126.
Cf. also A291761, A300229.
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, Mar 01 2018
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STATUS
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approved
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0, 2, 5, 7, 9, 16, 14, 29, 12, 16, 20, 67, 27, 16, 23, 121, 35, 67, 44, 67, 23, 16, 54, 277, 18, 16, 38, 67, 65, 436, 77, 497, 23, 16, 31, 631, 90, 16, 23, 277, 104, 436, 119, 67, 80, 16, 135, 1129, 25, 67, 23, 67, 152, 277, 31, 277, 23, 16, 170, 1771, 189, 16, 80, 2017, 31, 436, 209, 67, 23, 436, 230, 2557, 252, 16, 80, 67, 40, 436, 275, 1129, 138, 16, 299
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OFFSET
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1,2
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function
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FORMULA
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a(n) = (1/2)*(2 + ((A055396(n)+A046523(n))^2) - A055396(n) - 3*A046523(n)).
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PROG
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(PARI)
A001511(n) = (1+valuation(n, 2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286164(n) = (2 + ((A055396(n)+A046523(n))^2) - A055396(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286164.txt", n, " ", A286164(n)));
(Scheme) (define (A286164 n) (* (/ 1 2) (+ (expt (+ (A055396 n) (A046523 n)) 2) (- (A055396 n)) (- (* 3 (A046523 n))) 2)))
(Python)
from sympy import primepi, isprime, primefactors, factorint
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a046523(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1
def a(n): return T(a055396(n), a046523(n)) # Indranil Ghosh, May 05 2017
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CROSSREFS
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Cf. A000027, A046523, A055396, A286160, A286161, A286162, A286163.
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, May 04 2017
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STATUS
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approved
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A286360
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Compound filter (prime signature & sum of the divisors): a(n) = P(A046523(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.
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+20
14
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1, 8, 12, 49, 23, 142, 38, 239, 124, 259, 80, 753, 107, 412, 412, 1051, 173, 1237, 212, 1390, 672, 826, 302, 3427, 565, 1087, 1089, 2223, 467, 5080, 530, 4403, 1384, 1717, 1384, 7911, 743, 2086, 1836, 6352, 905, 7780, 992, 4477, 3928, 2932, 1178, 14583, 1774, 5368, 2932, 5898, 1487, 10177, 2932, 10177, 3576, 4471, 1832, 25711, 1955, 5056, 6567, 18019, 3922
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OFFSET
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1,2
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function
Index entries for sequences related to sigma(n)
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FORMULA
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a(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n)).
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PROG
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(PARI)
A000203(n) = sigma(n);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A286360(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n));
for(n=1, 10000, write("b286360.txt", n, " ", A286360(n)));
(Scheme) (define (A286360 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A000203 n)) 2) (- (A046523 n)) (- (* 3 (A000203 n))) 2)))
(Python)
from sympy import factorint, divisor_sigma as D
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a046523(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1
def a(n): return T(a046523(n), D(n)) # Indranil Ghosh, May 12 2017
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CROSSREFS
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Cf. A000027, A286359, A286460.
Cf. A007503, A065608 (sequences matching to this filter), also A000203, A046523, A161942, A286034, A286357.
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, May 10 2017
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STATUS
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approved
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 44, 49, 50, 51, 44, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 58, 62, 65, 66, 67, 68, 69, 70, 58, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 80
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refs;
listen;
history;
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internal format)
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OFFSET
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1,2
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COMMENTS
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Restricted growth sequence transform of A291752.
For all i, j:
a(i) = a(j) => A291751(i) = A291751(j),
a(i) = a(j) => A326199(i) = A326199(j) => A294877(i) = A294877(j),
a(i) = a(j) => A322021(i) = A322021(j),
a(i) = a(j) => A295888(i) = A295888(j),
a(i) = a(j) => A296090(i) = A296090(j).
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..100000
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PROG
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(PARI)
up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux295300(n) = (1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n));
v295300 = rgs_transform(vector(up_to, n, Aux295300(n)));
A295300(n) = v295300[n];
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CROSSREFS
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Cf. A003557, A046523, A048250, A101296, A286360, A291750, A291751, A291752, A291757, A291758, A295888, A296090, A322021, A326199.
Cf. also A305801, A305900, A323368, A324400.
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, Nov 19 2017
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EXTENSIONS
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Name changed and the comments section added by Antti Karttunen, Jul 13 2019
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STATUS
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approved
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