A071364 Smallest number with same sequence of exponents in canonical prime factorization as n.

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, 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, 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

Offset: 1

Author

Reinhard Zumkeller, May 21 2002

Keywords

• nonn

Comments

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);

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;

a(A003586(n))=A003586(n).

Examples

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

Links

• Daniel Forgues, Table of n, a(n) for n=1..100000

Crossrefs

Cf. A071365, A071366.

Cf. A000040.

Cf. A085079, A089247.

The range is A055932.

The reversed version is A331580.

Unsorted prime signature is A124010.

Numbers whose prime signature is aperiodic are A329139.

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

Programs

Haskell

a071364 = product . zipWith (^) a000040_list . a124010_row
-- Reinhard Zumkeller, Feb 19 2012

Mathematica

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

PARI

a(n) = f = factor(n); for (i=1, #f~, f[i,1] = prime(i)); factorback(f); \\ Michel Marcus, Jun 13 2014

Python

from math import prod
from sympy import prime, factorint
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

Formula

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

a(n) = product(A000040(k)^A124010(k): k=1..A001221(n)). - Reinhard Zumkeller, Apr 27 2013

Extensions

Extended by Ray Chandler, Sep 23 2005