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

1,2

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.

The largest exponential divisor (A322791) of n that is a number whose prime factorization exponents are all powers of 2 (A138302). The number of these divisors is A392447(n), and their sum is A392448(n). - Amiram Eldar, Jan 13 2026

Links

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

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A006519, A037445, A077609, A007947, A034444, A138302, A268335, A302792, A322791, A339597, A363329, A392447, A392448.

Sequence in context: A167514 A091733 A066990 * A097449 A104415 A071074

Adjacent sequences: A365293 A365294 A365295 * A365297 A365298 A365299

Keyword

nonn,easy,mult

Author

Amiram Eldar, Aug 31 2023

Status

approved