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A370329
a(n) is the number of coreful divisors of the n-th powerful number that are also powerful numbers.
2
1, 1, 2, 1, 3, 1, 2, 4, 1, 1, 5, 2, 3, 1, 2, 1, 2, 6, 3, 1, 1, 2, 4, 1, 4, 7, 4, 1, 3, 2, 1, 2, 3, 6, 1, 1, 2, 8, 1, 5, 3, 6, 2, 1, 5, 3, 4, 1, 8, 1, 1, 2, 4, 4, 9, 1, 2, 6, 1, 1, 9, 2, 2, 2, 1, 2, 1, 1, 4, 5, 1, 10, 1, 2, 1, 3, 8, 6, 3, 10, 1, 6, 2, 1, 7, 2, 3
OFFSET
1,3
COMMENTS
A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).
The positive terms of A361430.
LINKS
FORMULA
a(n) = A361430(A001694(n)).
a(n) = 1 if and only if n is the square of a squarefree number (A062503).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3/2) * zeta(3) * Product_{p prime} (1 + 2/p^2 + 2/p^(5/2) - 1/p^3 - 2/p^(7/2) - 2/p^4) = 6.91748056612108993003... . (The infinite product of primes is the value of f(1/2) in A361430).
MATHEMATICA
f[p_, e_] := e - 1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{max = 10^4}, s /@ Union@ Flatten@ Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]
PROG
(PARI) lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[, 2]; if(k == 1 || vecmin(e) > 1, print1(prod(i = 1, #e, e[i]-1), ", "))); }
CROSSREFS
Cf. A001694, A002117, A062503, A078434, A307958, A360908 (analogous with squares), A361430, A370328.
Sequence in context: A228812 A341049 A144113 * A304038 A366113 A301983
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Feb 15 2024
STATUS
approved