A339667 Number of nonempty subsets of divisors of n having a common factor > 1.

0, 1, 1, 3, 1, 5, 1, 7, 3, 5, 1, 19, 1, 5, 5, 15, 1, 19, 1, 19, 5, 5, 1, 71, 3, 5, 7, 19, 1, 37, 1, 31, 5, 5, 5, 111, 1, 5, 5, 71, 1, 37, 1, 19, 19, 5, 1, 271, 3, 19, 5, 19, 1, 71, 5, 71, 5, 5, 1, 347, 1, 5, 19, 63, 5, 37, 1, 19, 5, 37, 1, 703, 1, 5, 19, 19, 5, 37, 1, 271

Offset: 1

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

• nonn

Examples

			a(12) = 19 subsets: {2}, {3}, {4}, {6}, {12}, {2, 4}, {2, 6}, {2, 12}, {3, 6}, {3, 12}, {4, 6}, {4, 12}, {6, 12}, {2, 4, 6}, {2, 4, 12}, {2, 6, 12}, {3, 6, 12}, {4, 6, 12} and {2, 4, 6, 12}.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to divisors of numbers

Crossrefs

Cf. A000005, A008683, A027750, A076078, A100587, A109511.

Programs

Mathematica

Table[-DivisorSum[n, MoebiusMu[n/#] (2^DivisorSigma[0, #] - 1) &, # < n &], {n, 80}]

PARI

A339667(n) = -sumdiv(n, d, if(d==n,0, moebius(n/d)*((2^numdiv(d))-1))); \\ Antti Karttunen, Dec 15 2021

Formula

a(n) = -Sum_{d|n, d < n} mu(n/d) * (2^tau(d) - 1), where tau = A000005, and mu = A008683.

a(n) = A100587(n) - A076078(n).

a(p) = 1 for p prime.