Search: keyword:new
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A371919
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Number of almost controllable simple graphs on n vertices.
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+0
0
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OFFSET
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1,2
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LINKS
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CROSSREFS
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Cf. A356669 (numbers of controllable graphs).
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KEYWORD
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nonn,more,new
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AUTHOR
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STATUS
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approved
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A371696
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Composite numbers that divide the reverse of the concatenation of their ascending order prime factors, with repetition.
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+0
0
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26, 38, 46, 378, 26579, 84941, 178838, 30791466, 39373022, 56405502, 227501395, 904085931
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OFFSET
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1,1
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LINKS
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EXAMPLE
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26 is a term as 26 = 2 * 13 = "213" which is reverse is "312", and 312 is divisible by 26.
227501395 is a term as 227501395 = 5 * 11 * 17 * 23 * 71 * 149 = "511172371149" which in reverse is "941173271115", and 941173271115 is divisible by 227501395.
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PROG
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(Python)
from itertools import count, islice
from sympy import isprime, factorint
def ok(k): return not isprime(k) and int("".join(str(p)[::-1]*e for p, e in list(factorint(k).items())[::-1]))%k == 0
def agen(): yield from filter(ok, count(4))
print(list(islice(agen(), 7))) # Michael S. Branicky, Apr 13 2024
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CROSSREFS
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KEYWORD
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nonn,base,more,new
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AUTHOR
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STATUS
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approved
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A371923
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Lateral surface area of a right circular conoid with unit height and base radius.
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+0
0
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6, 0, 2, 7, 2, 1, 2, 5, 3, 4, 5, 4, 7, 3, 3, 3, 0, 3, 3, 4, 0, 7, 1, 7, 3, 5, 0, 4, 1, 4, 5, 6, 7, 9, 9, 2, 7, 1, 0, 3, 7, 5, 7, 4, 3, 0, 0, 3, 1, 4, 1, 0, 5, 7, 4, 7, 7, 1, 8, 9, 2, 6, 9, 9, 7, 2, 8, 9, 0, 8, 0, 1, 4, 0, 3, 9, 8, 8, 8, 2
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OFFSET
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1,1
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LINKS
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EXAMPLE
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6.02721253454...
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MATHEMATICA
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RealDigits[Pi HypergeometricPFQ[{-1/2, 1/4, 3/4}, {1/2, 1}, -1] - NIntegrate[(Sqrt[u] (1 + u) ArcTanh[Sqrt[(1 - u)/(1 + u^2)]])/(u - 1), {u, 0, 1}, WorkingPrecision -> 250, PrecisionGoal -> 120]][[1]]
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KEYWORD
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AUTHOR
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STATUS
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approved
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A371781
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Numbers with biquanimous prime signature.
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+0
6
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1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159
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OFFSET
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1,2
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COMMENTS
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First differs from A320911 in lacking 900.
First differs from A325259 in having 1 and lacking 120.
A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 (aerated) and ranked by A357976.
Also numbers n with a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.
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LINKS
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EXAMPLE
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The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is not in the sequence.
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MATHEMATICA
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g[n_]:=Select[Divisors[n], GCD[#, n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
Select[Range[100], g[#]!={}&]
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CROSSREFS
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A number's prime signature is given by A124010.
Partitions of this type are counted by A371839.
A371783 counts k-quanimous partitions.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A371782
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Numbers with non-biquanimous prime signature.
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+0
6
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2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 102
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OFFSET
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1,1
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COMMENTS
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A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 (aerated) and ranked by A357976.
Also numbers n without a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.
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LINKS
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EXAMPLE
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The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is in the sequence.
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MATHEMATICA
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g[n_]:=Select[Divisors[n], GCD[#, n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
Select[Range[100], g[#]=={}&]
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CROSSREFS
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A number's prime signature is given by A124010.
The complement for prime indices is A357976, counted by A002219 aerated.
Partitions of this type are counted by A371840.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A371926
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Number of Dyck paths of semilength n with strongly unimodal peak heights such that neighbouring peaks differ in height by exactly one and first and last peak are at height one.
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+0
0
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1, 1, 0, 0, 1, 0, 0, 1, 2, 1, 1, 4, 8, 11, 14, 23, 44, 79, 130, 209, 347, 598, 1042, 1801, 3084, 5273, 9060, 15658, 27152, 47122, 81769, 141919, 246525, 428742, 746479, 1300806, 2268169, 3956840, 6905817, 12057999, 21063319, 36809385, 64350631, 112535774
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OFFSET
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0,9
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LINKS
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EXAMPLE
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a(7) = 1: /\
/\/ \/\
/\/ \/\
a(8) = 2: /\ /\
/\/ \ /\ /\ / \/\
/\/ \/ \/\ /\/ \/ \/\
a(9) = 1: /\
/\ / \ /\
/\/ \/ \/ \/\
a(10) = 1: /\
/\/ \/\
/\/ \/\
/\/ \/\ .
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MAPLE
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b:= proc(x, y, v) option remember; (t-> `if`(x=t, 1,
`if`(x<t, 0, add(b(x-1-2*i, y-1, 0), i=1..y-1)+
`if`(v=1, add(b(x-1-2*i, y+1, v), i=1..y), 0))))(3*y-2)
end:
a:= n-> `if`(n=0, 1, b(2*n-1, 1$2)):
seq(a(n), n=0..50);
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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1, 12, 24, 90, 204, 330, 540, 1080, 1140, 2184, 3480, 6324, 15630, 23496, 38340, 48510, 56760, 99636, 234960, 270180, 300150, 528180, 703080, 973644, 1907178, 5992380, 7980930, 12032640, 20687436, 23847642, 27465840, 28653720, 34964340
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OFFSET
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1,2
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COMMENTS
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The corresponding record values are 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, ... (see the link for more values).
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LINKS
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EXAMPLE
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The iterations of A033880 over the first 8 terms:
n | a(n) | Iterations
--+------+----------------------------------------------------------------
1 | 1 | 1 -> -1
2 | 12 | 12 -> 4 -> -1
3 | 24 | 24 -> 12 -> 4 -> -1
4 | 90 | 90 -> 54 -> 12 -> 4 -> -1
5 | 204 | 204 -> 96 -> 60 -> 48 -> 28 -> 0
6 | 330 | 330 -> 204 -> 96 -> 60 -> 48 -> 28 -> 0
7 | 540 | 540 -> 600 -> 660 -> 696 -> 408 -> 264 -> 192 -> 124 -> -24
8 | 1080 | 1080 -> 1440 -> 2034 -> 378 -> 204 -> 96 -> 60 -> 48 -> 28 -> 0
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MATHEMATICA
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ab[n_] := Module[{k}, If[n < 1, 0, k = DivisorSigma[1, n] - 2*n; If[k < 1, 0, k]]]; f[n_] := Module[{s = NestWhileList[ab, n, UnsameQ, All]}, If[s[[-1]] == 0, Length[s] - 2, 0]]; seq[max_] := Module[{fm = 0, f1, s = {}}, Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, max}]; s]; seq[10^5]
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PROG
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(PARI) ab(n) = {my(k); if(n < 1, 0, k = sigma(n) - 2*n; if(k < 1, 0, k)); }
f(n) = {my(t = 0); until(bittest(t, n = ab(n)), t += 1<<n); if(n == 0, hammingweight(t) - 1, 0); } \\ after M. F. Hasler at A098007
lista(kmax) = {my(fm = 0, f1); for(k = 1, kmax, f1 = f(k); if(f1 > fm, fm = f1; print1(k, ", "))); }
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CROSSREFS
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KEYWORD
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nonn,more,new
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AUTHOR
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STATUS
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approved
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A371921
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The number of iterations of the map x -> A033880(x) starting at n until the a nonpositive number is reached, or 0 if this does not happen.
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+0
0
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1
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OFFSET
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1,12
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COMMENTS
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LINKS
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FORMULA
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a(n) = 1 if and only if n is nonabundant (A263837).
If a(n) > 0 then:
a(n) > 1 if n is abundant (A005101).
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EXAMPLE
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a(n) = 0 if the iterations that start at n are entering a cycle. Examples of cycles are:
1) Cycles of length 1: the triperfect numbers (A005820), 120, 672, 523776, ..., which are the fixed points of A033880. The triperfect numbers can be reached from other values of n, e.g., 276, 448, 486, 510, 702, ... .
2) Cycles of length 2: the only known cycle is (45840, 51168) (see A069085). It can be reached from other values of n, e.g., 32130, 39420, 45480, 66300, ... .
3) Cycles of length 3: the least cycle is (243732672, 271303776, 256786848). It is first reached from n = 107689320.
4) Cycles of length 4: the least cycle is (65071776, 82842816, 89761152, 77260656). It can be reached from other values of n, e.g., 33623940, 41132280, 42825888, ... . The next cycle of length 4 is (985948800, 1381340160, 2183133696, 1489384608).
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MATHEMATICA
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ab[n_] := Module[{k}, If[n < 1, 0, k = DivisorSigma[1, n] - 2*n; If[k < 1, 0, k]]]; a[n_] := Module[{s = NestWhileList[ab, n, UnsameQ, All]}, If[s[[-1]] == 0, Length[s] - 2, 0]]; Array[a, 120]
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PROG
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(PARI) ab(n) = {my(k); if(n < 1, 0, k = sigma(n) - 2*n; if(k < 1, 0, k)); }
a(n) = {my(t = 0); until(bittest(t, n = ab(n)), t += 1<<n); if(n == 0, hammingweight(t) - 1, 0); } \\ after M. F. Hasler at A098007
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CROSSREFS
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KEYWORD
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nonn,easy,new
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AUTHOR
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STATUS
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approved
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A371920
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Abundant numbers whose abundance is also an abundant number.
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+0
0
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24, 30, 42, 54, 60, 66, 78, 84, 90, 96, 102, 112, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 176, 180, 186, 198, 204, 208, 210, 216, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282, 294, 304, 306, 308, 312, 318, 330, 336, 342, 348, 354, 360
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OFFSET
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1,1
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COMMENTS
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First differs from A125639 at n = 12.
This sequence is infinite: if m is divisible by 6 and coprime to 5, then 5*m is a term.
All the multiply-perfect numbers (A007691) that are not 1 or perfect (A000396), i.e., the terms of A166069, are terms of this sequence.
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LINKS
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EXAMPLE
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MATHEMATICA
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ab[n_] := DivisorSigma[1, n] - 2*n; q[n_] := Module[{k = ab[n]}, k > 0 && ab[k] > 0]; Select[Range[360], q]
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PROG
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(PARI) ab(n) = sigma(n) - 2*n;
is(n) = {my(k = ab(n)); k > 0 && ab(k) > 0; }
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CROSSREFS
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KEYWORD
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nonn,easy,new
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AUTHOR
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STATUS
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approved
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A371913
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G.f. A(x) satisfies A(x) = 1 - x/A(x)^4 * (1 - A(x) - A(x)^5).
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+0
0
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1, 1, 2, 0, -6, 12, 67, -152, -740, 2296, 9017, -35979, -113936, 579516, 1454975, -9493390, -18317155, 157178640, 220172289, -2618995381, -2377680689, 43783556265, 19149194005, -732638868460, 16196837316, 12246524817736, -5891297294673
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n-5*k,n-k-1) for n > 0.
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PROG
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(PARI) a(n) = if(n==0, 1, sum(k=0, n, binomial(n, k)*binomial(n-5*k, n-k-1))/n);
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CROSSREFS
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KEYWORD
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sign,new
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AUTHOR
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STATUS
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approved
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