A345096 a(n) = Sum_{k=1..n} Sum_{i=1..k} k^(1 - ceiling(n/k^i) + floor(n/k^i)).

1, 4, 8, 15, 19, 29, 34, 48, 57, 69, 76, 101, 103, 125, 140, 166, 169, 206, 208, 247, 259, 285, 298, 353, 357, 389, 418, 457, 463, 529, 526, 589, 605, 645, 674, 756, 739, 797, 832, 903, 901, 991, 988, 1069, 1109, 1149, 1174, 1294, 1285, 1366, 1394, 1471, 1483, 1601, 1608

Offset

1,2

Comments

If p is prime, a(p) = Sum_{k=1..p} Sum_{i=1..k} k^(1 - ceiling(p/k^i) + floor(p/k^i)) = 1*(1^1) + ( 2*(2)^0 + ... + (p-1)*(p-1)^0 ) + p^1 + (p-1)(p^0) = p*(p-1)/2 + p + (p-1) = (p^2 + 3*p - 2)/2.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Vaclav Kotesovec, Plot of a(n)/(n^2/2) for n = 1..1000

Formula

Conjecture: a(n) ~ n^2/2. - Vaclav Kotesovec, Jun 08 2021

Example

a(4) = Sum_{k=1..4} Sum_{i=1..k} k^(1 - ceiling(4/k^i) + floor(4/k^i)) = (1^1) + (2^1 + 2^1) + (3^0 + 3^0 + 3^0) + (4^1 + 4^0 + 4^0 + 4^0) = 15.

Mathematica

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

Crossrefs

Sequence in context: A312742 A312743 A267368 * A190692 A309322 A312744

Adjacent sequences: A345093 A345094 A345095 * A345097 A345098 A345099

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 07 2021

Status

approved