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A173382 Partial sums of A074206. 3

%I

%S 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,

%T 90,93,97,105,106,119,120,136,139,142,145,171,172,175,178,198,199,212,

%U 213,221,229,232,233,281,283,291,294,302,303,323,326,346,349,352,353,397,398,401,409,441,444,457

%N Partial sums of A074206.

%C Partial sums of number of ordered factorizations of n.

%D 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.

%D 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.

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

%H N. J. A. Sloane, <a href="/A173382/b173382.txt">Table of n, a(n) for n = 0..20000</a>

%H Vaclav Kotesovec, <a href="/A173382/a173382.jpg">Graph - the asymptotic ratio</a>

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

%F 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

%e 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.

%t 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 *)

%t 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 *)

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

%Y A025523 is an essentially identical sequence.

%K easy,nonn

%O 0,3

%A _Jonathan Vos Post_, Feb 17 2010

%E Terms corrected by _N. J. A. Sloane_, May 04 2016

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Last modified September 18 22:48 EDT 2020. Contains 337174 sequences. (Running on oeis4.)