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A046523
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Smallest number with same prime signature as n.
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424
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1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 12, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, 12, 6, 30, 2, 72, 2, 6, 12, 12, 6, 30, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2
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internal format)
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OFFSET
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1,2
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
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FORMULA
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In prime factorization of n, replace most common prime by 2, next most common by 3, etc.
a(n) = A124859(A124859(n)) = A181822(A124859(n)). - Matthew Vandermast, May 19 2012
a(n) = A181821(A181819(n)). - Alois P. Heinz, Feb 17 2020
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EXAMPLE
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If p,q,... are different primes, a(p)=2, a(p^2)=4, a(pq)=6, a(p^2*q)=12, etc.
n = 108 = 2*2*3*3*3 is replaced by a(n) = 2*2*2*3*3 = 72;
n = 105875 = 5*5*5*7*11*11 is represented by a(n) = 2*2*2*3*3*5 = 360.
Prime-powers are replaced by corresponding powers of 2, primes by 2.
Factorials, primorials and lcm[1..n] are in the sequence.
A000005(a(n)) = A000005(n) remains invariant; least and largest prime factors of a(n) are 2 or p[A001221(n)] resp.
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MAPLE
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a:= n-> (l-> mul(ithprime(i)^l[i][2], i=1..nops(l)))
(sort(ifactors(n)[2], (x, y)->x[2]>y[2])):
seq(a(n), n=1..100); # Alois P. Heinz, Aug 18 2014
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MATHEMATICA
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Table[Apply[Times, p[w]^Reverse[Sort[ex[w]]]], {w, 1, 1000}] p[x_] := Table[Prime[w], {w, 1, lf[x]}] ex[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]]
ps[n_] := Sort[Last /@ FactorInteger[n]]; Join[{1}, Table[i = 2; While[ps[n] != ps[i], i++]; i, {n, 2, 89}]] (* Jayanta Basu, Jun 27 2013 *)
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PROG
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(PARI) a(n)=my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ Charles R Greathouse IV, Aug 17 2011
(PARI) A046523(n)=factorback(primes(#n=vecsort(factor(n)[, 2], , 4)), n) \\ M. F. Hasler, Oct 12 2018, improved Jul 18 2019
(Haskell)
import Data.List (sort)
a046523 = product .
zipWith (^) a000040_list . reverse . sort . a124010_row
-- Reinhard Zumkeller, Apr 27 2013
(Python)
from sympy import factorint
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1 # Indranil Ghosh, May 05 2017
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CROSSREFS
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A025487 gives range of values of this sequence.
Cf. A000142, A002110, A003418, A001221, A000040, A000005, A124010, A071364, A085079, A089247.
Cf. A181819, A181821.
Sequence in context: A083260 A284011 A275468 * A278524 A278523 A071364
Adjacent sequences: A046520 A046521 A046522 * A046524 A046525 A046526
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Corrected and extended by Ray Chandler, Mar 11 2004
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STATUS
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approved
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