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A359411 a(n) is the number of divisors of n that are both infinitary and exponential.

%I A359411 #15 Sep 01 2023 11:26:46
%S A359411 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,
%T A359411 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,
%U A359411 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A359411 a(n) is the number of divisors of n that are both infinitary and exponential.
%C A359411 First differs from A318672 and A325989 at n = 32.
%C A359411 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 an infinitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e and the bitwise AND of e and k is equal to k.
%C A359411 The least term that is higher than 2 is a(216) = 4.
%C A359411 The position of the first appearance of a prime p in this sequence is 2^A359081(p), if A359081(p) > -1. E.g., 2^39 = 549755813888 for p = 3, 2^175 = 4.789...*10^52 for p = 5, and 2^1275 = 6.504...*10^383 for p = 7.
%C A359411 This sequence is unbounded since A246600 is unbounded (see A359082).
%H A359411 Amiram Eldar, <a href="/A359411/b359411.txt">Table of n, a(n) for n = 1..10000</a>
%H A359411 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A359411 Multiplicative with a(p^e) = A246600(e).
%F A359411 a(n) = 1 if and only if n is in A138302.
%F A359411 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A246600(k)/p^k) = 1.135514937... .
%e A359411 a(8) = 2 since 8 has 2 divisors that are both infinitary and exponential: 2 and 8.
%t A359411 s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A359411 (PARI) s(n) = sumdiv(n, d, bitand(d, n)==d);
%o A359411 a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i,2]));}
%o A359411 (Python)
%o A359411 from math import prod
%o A359411 from sympy import divisors, factorint
%o A359411 def A359411(n): return prod(sum(1 for d in divisors(e,generator=True) if e|d == e) for e in factorint(n).values()) # _Chai Wah Wu_, Sep 01 2023
%Y A359411 Cf. A037445, A049419, A077609, A080948, A138302, A246600, A322791, A359081, A359082, A359412.
%Y A359411 Cf. A318672, A325989.
%K A359411 nonn,mult
%O A359411 1,8
%A A359411 _Amiram Eldar_, Dec 30 2022