A344622 a(n) = n*(n+1)/2 - sigma(n) + d(n).

1, 2, 4, 6, 11, 13, 22, 25, 35, 41, 56, 56, 79, 85, 100, 110, 137, 138, 172, 174, 203, 221, 254, 248, 297, 313, 342, 356, 407, 401, 466, 471, 517, 545, 586, 584, 667, 685, 728, 738, 821, 815, 904, 912, 963, 1013, 1082, 1062, 1171, 1188, 1258, 1286, 1379, 1373, 1472, 1484

Offset

1,2

Comments

For 1 <= k <= n, add 1 if k|n, otherwise add k. For example, a(12) = 1 + 1 + 1 + 1 + 5 + 1 + 7 + 8 + 9 + 10 + 11 + 1 = 56.

If p is prime, a(p) = p*(p+1)/2 - sigma(p) + d(p) = p*(p+1)/2 - (p+1) + 2 = 1 + p*(p-1)/2.

Links

Table of n, a(n) for n=1..56.

Formula

a(n) = Sum_{k=1..n} k^(ceiling(n/k)-floor(n/k)).

Example

a(10) = 10*(10+1)/2 - sigma(10) + d(10) = 55 - 18 + 4 = 41.

Mathematica

Table[n (n + 1)/2 - DivisorSigma[1, n] + DivisorSigma[0, n], {n, 80}]

Prog

(PARI) a(n) = n*(n+1)/2 - sigma(n) + numdiv(n); \\ Michel Marcus, May 25 2021

Crossrefs

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

Sequence in context: A350668 A078198 A171865 * A068059 A108868 A274261

Adjacent sequences: A344619 A344620 A344621 * A344623 A344624 A344625

Keyword

nonn

Author

Wesley Ivan Hurt, May 24 2021

Status

approved