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The smallest coreful infinitary divisor of n.
5

%I #19 Oct 21 2023 01:30:56

%S 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,

%T 3,28,29,30,31,2,33,34,35,36,37,38,39,10,41,42,43,44,45,46,47,48,49,

%U 50,51,52,53,6,55,14,57,58,59,60,61,62,63,4,65,66,67,68,69

%N The smallest coreful infinitary divisor of n.

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

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

%C All the terms are in A138302.

%H Amiram Eldar, <a href="/A365296/b365296.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F 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).

%F A037445(a(n)) = A034444(n). - _Amiram Eldar_, Oct 19 2023

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

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

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A365296(n): return prod(p**(e&-e) for p, e in factorint(n).items()) # _Chai Wah Wu_, Sep 01 2023

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

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Aug 31 2023