A330856 Total sum of divisors of all the parts in the partitions of n into 2 parts.

0, 2, 4, 11, 15, 25, 33, 48, 56, 75, 87, 111, 127, 149, 165, 204, 220, 251, 277, 315, 339, 383, 407, 459, 491, 536, 564, 628, 660, 714, 762, 825, 857, 923, 959, 1046, 1098, 1156, 1196, 1294, 1342, 1416, 1480, 1560, 1608, 1710, 1758, 1866, 1930, 2018, 2080, 2194, 2250

Offset

1,2

Links

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

Index entries for sequences related to partitions.

Formula

a(n) = Sum_{i=1..floor(n/2)} sigma(i) + sigma(n-i), where sigma(n) is the sum of divisors of n (A000203).

a(n) = ((n+1) mod 2) * sigma(floor(n/2)) + Sum_{i=1..n-1} sigma(i), where sigma(n) is the sum of divisors of n (A000203).

a(n) = A024916(n-1) for odd n >= 3, a(n) = A024916(n-1) + A000203(n/2) for even n. - Amiram Eldar, Dec 27 2024

Example

a(4) = 11; 4 has two partitions into 2 parts, (3,1) and (2,2). The total sum of all divisors of the parts is sigma(3) + sigma(1) + sigma(2) + sigma(2) = 4 + 1 + 3 + 3 = 11.

Maple

N:= 100: # for a(1) ... a(N)
S:= map(numtheory:-sigma, [$1..N]):
T:= ListTools:-PartialSums(S):
[0, seq(T[i-1]+`if`(i::even, S[i/2], 0), i=2..N)]; # Robert Israel, Apr 29 2020

Mathematica

Table[Sum[DivisorSigma[1, i] + DivisorSigma[1, n - i], {i, Floor[n/2]}], {n, 80}]

Crossrefs

Cf. A000203, A024916, A330857 (distinct parts).

Sequence in context: A214429 A002382 A356478 * A180384 A023168 A134419

Adjacent sequences: A330853 A330854 A330855 * A330857 A330858 A330859

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 27 2020

Status

approved