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

1,3

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

Links

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

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).

Example

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.

Mathematica

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

Prog

(Magma) [#[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]]
(PARI) a(n) = sumdiv(n, d, !(sigma(d) % numdiv(d))); \\ Michel Marcus, Oct 15 2020

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).

Sequence in context: A059129 A349954 A081771 * A066856 A089280 A246960

Adjacent sequences: A338167 A338168 A338169 * A338171 A338172 A338173

Keyword

nonn

Author

Jaroslav Krizek, Oct 14 2020

Extensions

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

Status

approved