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 A073642 Replace 2^k in the binary representation of n with k (i.e., if n = 2^a + 2^b + 2^c + ... then a(n) = a + b + c + ...). 13
 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 5, 5, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 15, 15, 6, 6, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 12, 12, 10, 10, 11, 11, 12, 12, 13, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS For n >= 1, a(n) is the n-th row sum of the irregular triangle A133457. - Vladimir Shevelev, Dec 14 2015 For n >= 0, 2^a(n) is the number of partitions of n whose dimension (given by the hook-length formula) is an odd integer. See the Macdonald reference. - Arvind Ayyer, May 12 2016 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 Arvind Ayyer, Amritanshu Prasad, Steven Spallone, Odd partitions in Young's lattice, arXiv:1601.01776 [math.CO], 2016. Ian G. Macdonald, On the degrees of the irreducible representations of symmetric groups, Bulletin of the London Mathematical Society, 3(2):189-192, 1971. FORMULA It seems that for n > 10 a(n) < n/(2*log(n)) and that Sum_{k=1..n} a(k) is asymptotic to C*n*log(n)^2 with 1/2 > C > 0.47. a(1)=0, a(2n) = a(n) + e1(n), a(2n+1) = a(2n), where e1(n) = A000120(n). - Ralf Stephan, Jun 19 2003 If n = 2^log_2(n) then a(n) = log_2(n); otherwise, a(n) = log_2(n) + a(n-2^log_2(n)), where log_2=A000523. a(2*n+1) = a(2*n), as the least significant bit of n does not contribute to a(n). - Reinhard Zumkeller, Aug 17 2003, edited by A.H.M. Smeets, Aug 17 2019 a(n) = Sum_{k=0..A070939(n)-1} k*A030308(n,k). - Reinhard Zumkeller, Jun 01 2013 Conjecture: a(n) = (3*A011371(n) - Sum_{k=1..n} A007814(k)^2)/2 for n > 0. - Velin Yanev, Sep 09 2017 G.f.: (1/(1 - x)) * Sum_{k>=1} k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Aug 17 2019 From A.H.M. Smeets, Aug 17 2019: (Start) floor(log_2(n)) <= a(n) <= floor(log_2(n+2)*(log_2(n+2)-1)/2), n > 0. Lower bound: floor(log_2(n)) = a(n) for n = 2^m or n = 2^m+1, m >= 0. Upper bound: a(n) = floor(log_2(n+2)*(log_2(n+2)-1)/2) for n = 2^m-2 or n = 2^m-1, m >= 0. (End) EXAMPLE 9 = 2^3 + 2^0, hence a(9) = 3 + 0 = 3; 25 = 2^4 + 2^3 + 2^0, hence a(25) = 4 + 3 + 0 = 7. MAPLE A073642 := proc(n)         local bdgs ;         bdgs := convert(n, base, 2) ;         add( op(i, bdgs)*(i-1), i=1..nops(bdgs)) ; end proc: # R. J. Mathar, Nov 17 2011 MATHEMATICA Total[Flatten[Position[Rest[Reverse[IntegerDigits[#, 2]]], 1]]] & /@ Range[0, 87] (* Jayanta Basu, Jul 03 2013 *) PROG (PARI) a(n)=sum(i=1, length(binary(n)), component(binary(n), i)*(length(binary(n))-i)) (Haskell) a073642 = sum . zipWith (*) [0..] . a030308_row -- Reinhard Zumkeller, Jun 01 2013 (Python) def A073642(n):     a, i = 0, 0     while n > 0:         a, n, i = a+(n%2)*i, n//2, i+1     return a print([A073642(n) for n in range(30)]) # A.H.M. Smeets, Aug 17 2019 CROSSREFS a(n) = log_2(A059867(n)). a(n) = A029931(floor(n/2)). Cf. A059867, A087135, A000009, A087136, A272090. Sequence in context: A225644 A225633 A060960 * A262869 A108356 A239499 Adjacent sequences:  A073639 A073640 A073641 * A073643 A073644 A073645 KEYWORD easy,nonn AUTHOR Benoit Cloitre, Aug 29 2002 EXTENSIONS a(0)=0 and offset corrected by Philippe Deléham, Apr 20 2009 STATUS approved

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Last modified July 9 09:40 EDT 2020. Contains 335542 sequences. (Running on oeis4.)