A345127 Total sum of the distinct prime factors of s*t, for all positive integer pairs (s,t) such that s + t = n.

0, 0, 2, 5, 7, 10, 17, 22, 23, 25, 36, 40, 52, 59, 63, 76, 84, 87, 106, 112, 116, 132, 155, 159, 173, 176, 191, 187, 215, 208, 254, 271, 261, 285, 291, 301, 337, 356, 364, 378, 418, 407, 471, 483, 486, 513, 560, 564, 591, 575, 585, 609, 661, 664, 672, 688, 698, 738, 797, 776

Offset

1,3

Links

Table of n, a(n) for n=1..60.

Formula

a(n) = Sum_{k=1..floor(n/2)} sopf(k*(n-k)).

Example

a(7) = 17; The partitions of 7 into two positive integer parts are (6,1), (5,2), and (4,3). The sum of the distinct prime factors of 6*1, 5*2, and 4*3 are 5, 7, and 5 respectively. The total is then 5 + 7 + 5 = 17.

Mathematica

Table[Sum[Sum[k (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[i (n - i)/k] + Floor[i (n - i)/k]), {k, i (n - i)}], {i, Floor[n/2]}], {n, 80}]

Prog

(PARI) a(n) = sum(k=1, n\2, vecsum(factor(k*(n-k))[, 1])); \\ Michel Marcus, Oct 31 2022

Crossrefs

Cf. A008472 (sopf).

Sequence in context: A103871 A068817 A156779 * A226813 A355846 A362388

Adjacent sequences: A345124 A345125 A345126 * A345128 A345129 A345130

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 08 2021

Status

approved