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A174605 Partial sums of A011371. 5
0, 0, 1, 2, 5, 8, 12, 16, 23, 30, 38, 46, 56, 66, 77, 88, 103, 118, 134, 150, 168, 186, 205, 224, 246, 268, 291, 314, 339, 364, 390, 416, 447, 478, 510, 542, 576, 610, 645, 680, 718, 756, 795, 834, 875, 916, 958, 1000, 1046, 1092, 1139, 1186, 1235, 1284, 1334 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Partial sums of n minus (number of 1's in binary expansion of n). Partial sums of highest power of 2 dividing n!. The subsequence of powers of 2 in the partial sum begins: 1, 2, 8, 16, no more through a(72) = 2414. The subsequence of primes in the partial sum begins: 2, 5, 23, 103, no more through a(72)= 2414. The subsequence of primes in the partial sum begins: 0, 1, 16, 576, no more through a(72)= 2414.

Exponent of 2 in the superfactorials, i.e., a(n) = A007814(A000178(n)). - Ralf Stephan, Jan 03 2014

REFERENCES

Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

LINKS

Table of n, a(n) for n=0..54.

A. Mir, F. Rossello and L. Rotger, A new balance index for phylogenetic trees, arXiv preprint arXiv:1202.1223, 2012

FORMULA

a(n) = SUM[i=0..n] A011371(i) = SUM[i=0..n] SUM[j=1..i](A007814(j), (i >= 1, a(0)=0) = SUM[i=0..n] (i - A000120(i)).

EXAMPLE

a(13) = 0 + 0 + 1 + 1 + 3 + 3 + 4 + 4 + 7 + 7 + 8 + 8 + 10 + 10 = 66.

MATHEMATICA

Accumulate[Table[n-DigitCount[n, 2, 1], {n, 0, 130}]] (* Harvey P. Dale, Feb 26 2015 *)

CROSSREFS

Cf. A011371, A000120, A005187, A054861, A032799, A067080, A098844, A132027.

Sequence in context: A219657 A213707 A229154 * A108577 A272719 A297834

Adjacent sequences:  A174602 A174603 A174604 * A174606 A174607 A174608

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Mar 23 2010

STATUS

approved

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Last modified August 6 00:46 EDT 2021. Contains 346493 sequences. (Running on oeis4.)