A344746 a(n) = Sum_{k=1..n} d(k) * k^c(n/k), where c(n) = 1 - ceiling(n) + floor(n).

1, 5, 9, 19, 18, 40, 28, 59, 51, 73, 49, 136, 61, 107, 113, 164, 84, 210, 96, 235, 166, 180, 120, 397, 167, 217, 227, 338, 159, 469, 173, 419, 275, 293, 287, 682, 214, 332, 330, 667, 240, 666, 254, 549, 538, 412, 280, 1056, 357, 619, 447, 658, 323, 907, 475, 944, 507, 533, 365

Offset: 1

Author

Wesley Ivan Hurt, Jun 02 2021

Keywords

• nonn

Comments

For 1 <= k <= n, if k|n then add k * d(k), otherwise add d(k).

If p is prime, a(p) = Sum_{k=1..p} d(k) * k^c(p/k) = 2*p + Sum_{k=1..p-1} d(k) = 2*p - 2 + d(p) + Sum_{k=1..p-1} d(k) = 2*p - 2 + Sum_{k=1..p} d(k).

Examples

			a(8) = Sum_{k=1..8} d(k) * k^c(8/k) = d(1)*1^1 + d(2)*2^1 + d(3)*3^0 + d(4)*4^1 + d(5)*5^0 + d(6)*6^0 + d(7)*7^0 + d(8)*8^1 = 1*1 + 2*2 + 2*1 + 3*4 + 2*1 + 4*1 + 2*1 + 4*8 = 59.
		

Crossrefs

Cf. A000005 (tau), A006218, A143127.

Programs

Mathematica

Table[Sum[DivisorSigma[0, k] k^(1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]