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 A359411 a(n) is the number of divisors of n that are both infinitary and exponential. 8
 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, 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, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS First differs from A318672 and A325989 at n = 32. 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. The least term that is higher than 2 is a(216) = 4. 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. This sequence is unbounded since A246600 is unbounded (see A359082). LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 Index entries for sequences computed from exponents in factorization of n. FORMULA Multiplicative with a(p^e) = A246600(e). a(n) = 1 if and only if n is in A138302. 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... . EXAMPLE a(8) = 2 since 8 has 2 divisors that are both infinitary and exponential: 2 and 8. MATHEMATICA s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] PROG (PARI) s(n) = sumdiv(n, d, bitand(d, n)==d); a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 2])); } (Python) from math import prod from sympy import divisors, factorint 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 CROSSREFS Cf. A037445, A049419, A077609, A080948, A138302, A246600, A322791, A359081, A359082, A359412. Cf. A318672, A325989. Sequence in context: A318672 A359910 A368168 * A367516 A368979 A325989 Adjacent sequences: A359408 A359409 A359410 * A359412 A359413 A359414 KEYWORD nonn,mult AUTHOR Amiram Eldar, Dec 30 2022 STATUS approved

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Last modified April 20 09:04 EDT 2024. Contains 371799 sequences. (Running on oeis4.)