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

Author

Wesley Ivan Hurt, Aug 21 2020

Keywords

• nonn
• easy
• changed

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

Examples

			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.
		

Crossrefs

Cf. A066446, A184389, A337246.

Cf. A000005, A000203, A337360.

Programs

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}]

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

Formula

a(n) = Sum_{d1|n, d2|n, d1

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

Extensions

New name using formula from Ridouane Oudra, Jul 31 2025

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