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

1,3

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.

Links

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

Formula

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Cf. A334420, A334421.

Sequence in context: A122662 A210228 A209161 * A333824 A083843 A094264

Adjacent sequences: A338168 A338169 A338170 * A338172 A338173 A338174

Keyword

nonn,look

Author

Jaroslav Krizek, Oct 14 2020

Status

approved