A365296 The smallest coreful infinitary divisor of n.

1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 6, 25, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 6, 55, 14, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69

Offset: 1

Author

Amiram Eldar, Aug 31 2023

Keywords

• nonn
• easy
• mult

Comments

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n.

The number of coreful infinitary divisors of n is A363329(n).

All the terms are in A138302.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006519, A037445, A077609, A007947, A034444, A138302, A268335, A302792, A339597, A363329.

Programs

Mathematica

f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2^valuation(f[i,2], 2)));}

Python

from math import prod
from sympy import factorint
def A365296(n): return prod(p**(e&-e) for p, e in factorint(n).items()) # Chai Wah Wu, Sep 01 2023

Formula

Multiplicative with a(p^e) = p^A006519(e).

a(n) = n if and only if n is in A138302.

a(n) >= A007947(n) with equality if and only if n is an exponentially odd number (A268335).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/p + Sum_{e>=1} 1/p^f(e)-1/p^(f(e)+1)) = 0.4459084041..., where f(k) = 2*k - A006519(k) = A339597(k-1).

A037445(a(n)) = A034444(n). - Amiram Eldar, Oct 19 2023