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A363329 a(n) is the number of divisors of n that are both coreful and infinitary.

%I A363329 #17 Mar 27 2025 05:43:27
%S A363329 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,3,1,1,1,1,3,1,1,
%T A363329 1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,3,1,1,1,1,
%U A363329 1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A363329 a(n) is the number of divisors of n that are both coreful and infinitary.
%C A363329 For the definition of a coreful divisor see A307958, and for the definition of an infinitary divisor see A037445.
%C A363329 If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a coreful and an infinitary divisor, the exponent of the highest power of p dividing d is a number k >= 1 such that the bitwise AND of e and k is equal to k.
%C A363329 The least term that does not equal 1 or 3 is a(128) = 7.
%C A363329 The range of this sequence is A282572.
%H A363329 Amiram Eldar, <a href="/A363329/b363329.txt">Table of n, a(n) for n = 1..10000</a>
%F A363329 Multiplicative with a(p^e) = 2^A000120(e) - 1.
%F A363329 a(n) = 1 is and only if n is in A138302.
%F A363329 a(n) >= A359411(n).
%F A363329 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (-1/p + (1-1/p)*Product_{k>=0} (1 + 2/p^(2^k))) = 1.29926464312956239535... .
%e A363329 a(8) = 3 since 8 has 4 divisors, 1, 2, 4 and 8, all are infinitary and 3 of them (2, 4 and 8) are also coreful.
%t A363329 f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A363329 (PARI) a(n) = factorback(apply(x -> 2^hammingweight(x) - 1, factor(n)[, 2]));
%o A363329 (Python)
%o A363329 from math import prod
%o A363329 from sympy import factorint
%o A363329 def A363329(n): return prod((1<<e.bit_count())-1 for e in factorint(n).values()) # _Chai Wah Wu_, Sep 01 2023
%Y A363329 Cf. A000120, A005361 (number of coreful divisors), A007947, A037445, A077609, A138302, A282572, A359411.
%K A363329 nonn,mult
%O A363329 1,8
%A A363329 _Amiram Eldar_, May 28 2023