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A173382
Partial sums of A074206.
4
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
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.
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
A025523 is an essentially identical sequence.
Sequence in context: A238617 A076061 A025523 * A128689 A116137 A178611
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 17 2010
EXTENSIONS
Terms corrected by N. J. A. Sloane, May 04 2016
STATUS
approved