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A365296
The smallest coreful infinitary divisor of n.
5
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.
LINKS
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
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Aug 31 2023
STATUS
approved