A337944 Total number of divisors of the sum of the squared parts in each partition of n into two parts.

0, 2, 2, 8, 4, 16, 7, 24, 14, 26, 14, 50, 18, 47, 24, 60, 23, 84, 30, 72, 45, 80, 37, 140, 36, 92, 64, 135, 46, 144, 53, 152, 84, 127, 59, 238, 62, 156, 100, 180, 71, 263, 76, 230, 110, 189, 85, 352, 102, 194, 133, 254, 98, 346, 98, 333, 162, 256, 114, 408, 118, 273, 201, 360

Offset

1,2

Links

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

Formula

a(n) = Sum_{k=1..floor(n/2)} tau(k^2+(n-k)^2), where tau(n) is the number of divisors of n (A000005).

a(n) = (tau(2*floor(n/2)^2)*mod(n+1,2)-tau(n^2)+Sum_{k=1..n} tau(k^2+(n-k)^2))/2, n>1. - Wesley Ivan Hurt, Jun 20 2026

Example

a(6) = 16; 6 has 3 partitions into two parts, (5,1), (4,2) and (3,3). The sums of the squared parts from each partition are 5^2 + 1^2 = 26, 4^2 + 2^2 = 20 and 3^2 + 3^2 = 18. Then d(26) + d(20) + d(18) = 4 + 4 + 8 = 16.

Maple

f:= proc(n) local i;
  add(NumberTheory:-tau(i^2+(n-i)^2), i=1..floor(n/2))
end proc:
map(f, [$1..100]); # Robert Israel, Jun 07 2026

Mathematica

Table[Sum[DivisorSigma[0, i^2 + (n - i)^2], {i, Floor[n/2]}], {n, 100}]

Prog

(PARI) a(n) = sum(i=1, n\2, numdiv(i^2+(n-i)^2)); \\ Michel Marcus, Nov 02 2022

Crossrefs

Cf. A000005 (tau), A337943.

Sequence in context: A144847 A143625 A003612 * A103839 A385185 A135727

Adjacent sequences: A337941 A337942 A337943 * A337945 A337946 A337947

Keyword

nonn,changed

Author

Wesley Ivan Hurt, Oct 01 2020

Status

approved