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 A006520 Partial sums of A006519. (Formerly M2344) 6

%I M2344

%S 1,3,4,8,9,11,12,20,21,23,24,28,29,31,32,48,49,51,52,56,57,59,60,68,

%T 69,71,72,76,77,79,80,112,113,115,116,120,121,123,124,132,133,135,136,

%U 140,141,143,144,160,161,163,164,168,169,171,172,180,181,183,184,188,189

%N Partial sums of A006519.

%C The subsequence of primes in this partial sum begins: 3, 11, 23, 29, 31, 59, 71, 79, 113, 163, 181. The subsequence of powers in this partial sum begins: 1, 4, 8, 9, 32, 49, 121, 144, 169. [_Jonathan Vos Post_, Feb 18 2010]

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A006520/b006520.txt">Table of n, a(n) for n = 1..1001</a> [Jul 23 2013; offset adapted by _Georg Fischer_, Jan 27 2020]

%H V. Meally, <a href="/A006516/a006516.pdf">Letter to N. J. A. Sloane, May 1975</a>

%H Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences ...</a>

%H Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>

%F a(n)/(n*log(n)) is bounded. - _Benoit Cloitre_, Dec 17 2002

%F G.f.: 1/x/(1-x) * (x/(1-x) + Sum(k>=1, 2^(k-1)*x^2^k/(1-x^2^k))). - _Ralf Stephan_, Apr 17 2003

%F a(n) = b(n+1), with b(2n) = 2b(n) + n, b(2n+1) = 2b(n) + n + 1. - _Ralf Stephan_, Sep 07 2003

%F a(2^k-1) = k*2^(k-1) = A001787(k) for any k > 0. - _Rémy Sigrist_, Jan 21 2021

%t Table[ 2^IntegerExponent[n, 2], {n, 1, 70}] // Accumulate (* _Jean-François Alcover_, May 14 2013 *)

%o (PARI) a(n)=sum(i=1,n,2^valuation(i,2))

%Y First differences of A022560.

%Y Cf. A001787.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from _Benoit Cloitre_, Dec 17 2002

%E Offset changed to 1 by _N. J. A. Sloane_, Oct 18 2019

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Last modified April 17 14:18 EDT 2021. Contains 343063 sequences. (Running on oeis4.)