A173382 Partial sums of A074206.

0, 1, 2, 3, 5, 6, 9, 10, 14, 16, 19, 20, 28, 29, 32, 35, 43, 44, 52, 53, 61, 64, 67, 68, 88, 90, 93, 97, 105, 106, 119, 120, 136, 139, 142, 145, 171, 172, 175, 178, 198, 199, 212, 213, 221, 229, 232, 233, 281, 283, 291, 294, 302, 303, 323, 326, 346, 349, 352, 353, 397, 398, 401, 409, 441, 444, 457

Offset

0,3

Comments

Partial sums of number of ordered factorizations of n.

References

Shikao Ikehara, On Kalmar's Problem in “Factorisatio Numerorum”, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 21 (1939) pp. 208-219.

Shikao Ikehara, On Kalmar's Problem in “Factorisatio Numerorum” II, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 23 (1941) pp. 767-774.

Kalmár, Laszlo. "Über die mittlere Anzahl der Produktdarstellungen der Zahlen.(Erste Mitteilung)'." Acta Litt. ac Scient. Szeged 5 (1931): 95-107.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Ann Clifton, Eva Czabarka, Kevin Liu, Sarah Loeb, Utku Okur, Laszlo Szekely, and Kristina Wicke, Universal rooted phylogenetic tree shapes and universal tanglegrams, arXiv:2308.06580 [math.CO], 2023.

Vaclav Kotesovec, Graph - the asymptotic ratio

Formula

a(n) = Sum_{i=0..n} A074206(i).

a(n) ~ -n^r / (r*Zeta'(r)), where r = A107311 = 1.728647238998183618135103... is the root of the equation Zeta(r) = 2. - Vaclav Kotesovec, Jan 31 2019

Example

a(96) = 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 2 + 3 + 1 + 8 + 1 + 3 + 3 + 8 + 1 + 8 + 1 + 8 + 3 + 3 + 1 + 20 + 2 + 3 + 4 + 8 + 1 + 13 + 1 + 16 + 3 + 3 + 3 + 26 + 1 + 3 + 3 + 20 + 1 + 13 + 1 + 8 + 8 + 3 + 1 + 48 + 2 + 8 + 3 + 8 + 1 + 20 + 3 + 20 + 3 + 3 + 1 + 44 + 1 + 3 + 8 + 32 + 3 + 13 + 1 + 8 + 3 + 13 + 1 + 76 + 1 + 3 + 8 + 8 + 3 + 13 + 1 + 48 + 8 + 3 + 1 + 44 + 3 + 3 + 3 + 20 + 1 + 44 + 3 + 8 + 3 + 3 + 3 + 112.

Mathematica

Clear[a]; a[0] = 0; a[1] = 1; a[n_] := a[n] = 1 + Sum[a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 0, 100}] (* Vaclav Kotesovec, Jan 31 2019 *)
Clear[a]; a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; Join[{0}, Accumulate[a /@ Range[100]]] (* Vaclav Kotesovec, Jan 31 2019, after Jean-François Alcover, faster *)

Crossrefs

Cf. A074206, A002033, A001055, A050324, A000670.

A025523 is an essentially identical sequence.

Sequence in context: A238617 A076061 A025523 * A128689 A116137 A178611

Adjacent sequences: A173379 A173380 A173381 * A173383 A173384 A173385

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 17 2010

Extensions

Terms corrected by N. J. A. Sloane, May 04 2016

Status

approved