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Smallest number with same sequence of exponents in canonical prime factorization as n.
28

%I #28 Sep 17 2022 02:00:29

%S 1,2,2,4,2,6,2,8,4,6,2,12,2,6,6,16,2,18,2,12,6,6,2,24,4,6,8,12,2,30,2,

%T 32,6,6,6,36,2,6,6,24,2,30,2,12,12,6,2,48,4,18,6,12,2,54,6,24,6,6,2,

%U 60,2,6,12,64,6,30,2,12,6,30,2,72,2,6,18,12,6,30,2,48,16,6,2,60,6,6,6,24

%N Smallest number with same sequence of exponents in canonical prime factorization as n.

%C A046523(a(n))=A046523(n); A046523(n)<=a(n)<=n; A001221(a(n))=A001221(n), A001222(a(n))=A001222(n); A020639(a(n))=2, A006530(a(n))=A000040(A001221(n))<=A006530(n); A000005(a(n))=A000005(n);

%C a(a(n))=a(n); a(n)=2^k iff n=p^k, p prime, k>0 (A000961); if n>1 is not a prime power, then a(n) mod 6 = 0; range of values = A055932, as distinct prime factors of a(n) are consecutive: a(n)=n iff n=A055932(k) for some k;

%C a(A003586(n))=A003586(n).

%H Daniel Forgues, <a href="/A071364/b071364.txt">Table of n, a(n) for n=1..100000</a>

%F In prime factorization of n, replace least prime by 2, next least by 3, etc.

%F a(n) = product(A000040(k)^A124010(k): k=1..A001221(n)). - _Reinhard Zumkeller_, Apr 27 2013

%e a(105875) = a(5*5*5*7*11*11) = 2*2*2*3*5*5 = 600.

%t Table[ e = Last /@ FactorInteger[n]; Product[Prime[i]^e[[i]], {i, Length[e]}], {n, 88}] (* _Ray Chandler_, Sep 23 2005 *)

%o (Haskell)

%o a071364 = product . zipWith (^) a000040_list . a124010_row

%o -- _Reinhard Zumkeller_, Feb 19 2012

%o (PARI) a(n) = f = factor(n); for (i=1, #f~, f[i,1] = prime(i)); factorback(f); \\ _Michel Marcus_, Jun 13 2014

%o (Python)

%o from math import prod

%o from sympy import prime, factorint

%o def A071364(n): return prod(prime(i+1)**p[1] for i,p in enumerate(sorted(factorint(n).items()))) # _Chai Wah Wu_, Sep 16 2022

%Y Cf. A071365, A071366.

%Y Cf. A000040.

%Y Cf. A085079, A089247.

%Y The range is A055932.

%Y The reversed version is A331580.

%Y Unsorted prime signature is A124010.

%Y Numbers whose prime signature is aperiodic are A329139.

%Y Cf. A056239, A112798, A233249, A333217, A333219, A334032, A334033.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, May 21 2002

%E Extended by _Ray Chandler_, Sep 23 2005