A337326 -id:A337326 - cp's OEIS frontend

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

Jaroslav Krizek, Oct 14 2020

nonn

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

			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.

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

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).

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

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

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

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).

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

A368215 a(n) is the smallest number k >= 1 that has exactly n divisors in A020487.

Original entry on oeis.org

1, 4, 16, 36, 256, 100, 200, 576, 400, 800, 2600, 900, 3200, 1800, 16900, 6400, 12800, 3600, 20800, 7200, 11700, 36000, 67600, 14400, 23400, 28800, 32400, 88200, 397800, 64800, 270400, 46800, 152100, 115200, 234000, 93600, 1258400, 230400, 259200, 352800, 1081600

Offset: 1

Marius A. Burtea, Dec 17 2023

nonn

a(n) exists for each n because 4^(n-1) has n antiharmonic divisors.

			a(1) = 1 because 1 has only one divisor 1 = A020487(1).
The numbers 2 and 3 have only the divisor 1 in A020487 and 4 has the divisors 1 = A020487(1) and 4 = A020487(2), so a(2) = 4.

Cf. A000005, A000203, A001157, A020487, A337326.

f:=func; a:=[]; for n in [1..41] do k:=1; while #[d:d in Divisors(k)|f(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

f[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[2, #], DivisorSigma[1, #]] &]; seq[len_] := Module[{s = Table[0, {len}], c = 0, n = 1}, While[c < len, If[(i = f[n]) <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[25] (* Amiram Eldar, Dec 18 2023 *)

a(n) = my(k=1); while(sumdiv(k, d, sigma(d, 2)%sigma(d)==0) != n, k++); k; \\ Michel Marcus, Dec 18 2023

cp's OEIS Frontend

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

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