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

1,2

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).

Links

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

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

Example

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

Mathematica

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

Prog

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

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.

Sequence in context: A046523 A278524 A278523 * A278237 A328707 A067824

Adjacent sequences: A071361 A071362 A071363 * A071365 A071366 A071367

Keyword

nonn

Author

Reinhard Zumkeller, May 21 2002

Extensions

Extended by Ray Chandler, Sep 23 2005

Status

approved