A338170 a(n) is the number of divisors d of n such that tau(d) divides sigma(d).

1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 2, 3, 3, 2, 6, 2, 1, 4, 2, 4, 3, 2, 3, 4, 3, 2, 7, 2, 4, 5, 3, 2, 3, 3, 2, 4, 2, 2, 5, 4, 4, 4, 2, 2, 8, 2, 3, 4, 1, 4, 7, 2, 3, 4, 6, 2, 3, 2, 2, 4, 3, 4, 6, 2, 3, 3, 2, 2, 7, 4, 3, 4, 4, 2, 7, 4, 4, 4, 3, 4, 4, 2, 4, 5, 3, 2, 6, 2, 2, 8

Offset: 1

Author

Jaroslav Krizek, Oct 14 2020

Keywords

• nonn

Comments

a(n) is the number of arithmetic divisors d of n.

a(n) = tau(n) = A000005(n) for numbers n from A334420.

See A338171 and A338172 for sum and product such divisors.

a(n) = 1 iff n = 2^k (A000079). - Bernard Schott, Dec 06 2020

Examples

			a(6) = 3 because there are 3 arithmetic divisors of 6 (1, 3 and 6):
sigma(1)/tau(1) =  1/1 = 1; sigma(3)/tau(3) = 4/2 = 2; sigma(6)/tau(6) = 12/4 = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Inverse Möbius transform of A245656.

Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).

Cf. A334420, A334421.

Cf. A337326 (smallest numbers m with n such divisors).

Programs

Magma

[#[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];

Mathematica

a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)

PARI

a(n) = sumdiv(n, d, !(sigma(d) % numdiv(d))); \\ Michel Marcus, Oct 15 2020

Formula

a(n) = Sum_{d|n} c(d), where c(n) is the arithmetic characteristic of n (A245656).

a(p) = 2 for odd primes p (A065091).

Extensions

Data section extended up to 105 terms by Antti Karttunen, Dec 12 2021