A338171 a(n) is the sum of those divisors d of n such that tau(d) divides sigma(d).

1, 1, 4, 1, 6, 10, 8, 1, 4, 6, 12, 10, 14, 22, 24, 1, 18, 10, 20, 26, 32, 34, 24, 10, 6, 14, 31, 22, 30, 60, 32, 1, 48, 18, 48, 10, 38, 58, 56, 26, 42, 94, 44, 78, 69, 70, 48, 10, 57, 6, 72, 14, 54, 91, 72, 78, 80, 30, 60, 140, 62, 94, 32, 1, 84, 142, 68, 86

Offset: 1

Author

Jaroslav Krizek, Oct 14 2020

Keywords

• nonn
• look

Comments

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

a(n) = sigma(n) = A000203(n) for numbers n from A334420.

See A338170 and A338172 for number and product such divisors.

Examples

			a(6) = 10 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. Sum of this divisors is 10.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

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

Cf. A334420, A334421.

Programs

Magma

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

Maple

f:= proc(n) uses numtheory;
convert(select(t -> sigma(t) mod tau(t) = 0, divisors(n)),`+`) end proc:
map(f, [$1..100]); # Robert Israel, Oct 27 2020

Mathematica

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

PARI

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

Formula

a(p) = p + 1 for odd primes p (A065091).