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A327513
Number of divisors of n that are 1, 2, or a nonprime number whose prime indices are pairwise coprime.
3
1, 2, 1, 3, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 2, 5, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 6, 1, 6, 2, 3, 2, 5, 1, 3, 1, 7, 1, 4, 1, 5, 2, 3, 1, 9, 1, 3, 2, 5, 1, 3, 2, 7, 1, 3, 1, 10, 1, 3, 1, 7, 1, 6, 1, 5, 2, 6, 1, 7, 1, 3, 2, 5, 2, 4, 1, 9, 1, 3, 1, 7, 2, 3, 1, 7, 1, 6, 1, 5, 2, 3, 2, 11, 1, 3, 2, 5, 1, 6, 1, 7, 3
OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Numbers that are 1, 2, or a nonprime number whose prime indices are pairwise coprime are listed in A302696.
Note that the maximum odd divisor of any entry must be squarefree.
Number of terms of A302696 that divide n. Put in other words, this sequence is the inverse Möbius transform of the characteristic function of A302696. - Antti Karttunen, Dec 06 2021
EXAMPLE
The divisors of 72 that are 1, 2, or nonprime numbers whose prime indices are pairwise coprime are: {1, 2, 4, 6, 8, 12, 24}, so a(72) = 7.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Divisors[n], #==1||CoprimeQ@@primeMS[#]&]], {n, 100}]
PROG
(PARI)
isA302696(n) = if(isprimepower(n), !(n%2), if(!issquarefree(n>>valuation(n, 2)), 0, my(pis=apply(primepi, factor(n)[, 1])); (lcm(pis)==factorback(pis))));
A327513(n) = sumdiv(n, d, isA302696(d)); \\ Antti Karttunen, Dec 06 2021
CROSSREFS
See link for additional cross-references.
Sequence in context: A326154 A306248 A361788 * A327530 A084360 A082460
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 19 2019
EXTENSIONS
Data section extended up to 105 terms by Antti Karttunen, Dec 06 2021
STATUS
approved