A337297 a(n) = sigma(n)*(tau(n) - 1).

0, 3, 4, 14, 6, 36, 8, 45, 26, 54, 12, 140, 14, 72, 72, 124, 18, 195, 20, 210, 96, 108, 24, 420, 62, 126, 120, 280, 30, 504, 32, 315, 144, 162, 144, 728, 38, 180, 168, 630, 42, 672, 44, 420, 390, 216, 48, 1116, 114, 465, 216, 490, 54, 840, 216, 840, 240, 270, 60, 1848, 62, 288

Offset

1,2

Comments

Original name was: Sum of the coordinates of all pairs of divisors of n, (d1,d2), such that d1 < d2.

If n = p where p is prime, the only pair of divisors of p such that d1 < d2 is (1,p), whose coordinate sum is a(p) = p + 1. - Wesley Ivan Hurt, May 21 2021

Links

Table of n, a(n) for n=1..62.

Formula

a(n) = Sum_{d1|n, d2|n, d1<d2} (d1 + d2).

a(p^k) = k*(p^(k+1)-1)/(p-1) for p prime and k >= 1. - Wesley Ivan Hurt, Aug 23 2025

a(n) = A064840(n) - A000203(n). - Wesley Ivan Hurt, Sep 21 2025

Example

a(3) = 4; The divisors of 3 are {1,3}. If we form all ordered pairs (d1,d2) such that d1 < d2, we have: (1,3). The sum of the coordinates gives 1+3 = 4.
a(4) = 14; The divisors of 4 are {1,2,4}. If we form all ordered pairs (d1,d2) such that d1<d2, we have: (1,2), (1,4), (2,4). The sum of all the coordinates gives 1+2+1+4+2+4 = 14.

Mathematica

Table[Sum[Sum[(i + k)*(1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}]
Table[DivisorSigma[1, n]*(DivisorSigma[0, n] - 1), {n, 60}] (* Wesley Ivan Hurt, Sep 21 2025 *)

Prog

(PARI) a(n) = my(d = divisors(n)); sum(i=1, #d, sum(j=1, i-1, d[i]+d[j])); \\ Michel Marcus, Aug 22 2020

Crossrefs

Cf. A064840, A066446, A184389, A337246.

Cf. A000005 (tau), A000203 (sigma), A337360.

Sequence in context: A216868 A082732 A307893 * A220846 A009286 A076663

Adjacent sequences: A337294 A337295 A337296 * A337298 A337299 A337300

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 21 2020

Extensions

New name using formula from Ridouane Oudra, Jul 31 2025

Status

approved