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A073119 Total number of parts which are positive powers of 2 in all partitions of n. 2
0, 1, 1, 4, 5, 10, 14, 26, 35, 56, 77, 116, 157, 226, 302, 424, 560, 762, 998, 1334, 1727, 2270, 2914, 3779, 4809, 6163, 7781, 9875, 12378, 15565, 19383, 24191, 29934, 37093, 45643, 56201, 68789, 84212, 102564, 124903, 151424, 183499, 221508 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Alois P. Heinz)

FORMULA

a(n) = Sum_{k=1..n} A007814(k)*A000041(n-k).

G.f.: g(x) = (Sum_{i>0} x^(h(i))/(1-x^(h(i))))/Product_{i>0}(1 - x^i), where h(i) = 2^i. - Emeric Deutsch, Sep 19 2016.

Conjecture: a(n) ~ exp(sqrt(2*n/3)*Pi)/(2*Pi*sqrt(2*n)) ~ p(n) * sqrt(6*n)/Pi, where p(n) is the partition function A000041. - Vaclav Kotesovec, Oct 07 2016

EXAMPLE

a(5) = 5 because in the partitions [1,1,1,1,1], [1,1,1,2'], [1,2'2'], [1,1,3], [2',3],[1,4'], [5] we have 5 positive powers of 2 (they are marked). - Emeric Deutsch, Sep 19 2016.

MAPLE

p2:= proc(n) p2(n):= is(n=2^ilog2(n)) end: p2(1):= false:

b:= proc(n, i) option remember; local t, l;

      if n<0 then [0, 0]

    elif n=0 then [1, 0]

    elif i<1 then [0, 0]

    else t:= b(n, i-1);

         l:= b(n-i, i);

         [t[1]+l[1], t[2]+l[2]+ `if`(p2(i), l[1], 0)]

      fi

    end:

a:= n-> b(n, n)[2]:

seq(a(n), n=1..50);  # Alois P. Heinz, Sep 29 2011

MATHEMATICA

Needs["DiscreteMath`Combinatorica`"]; f[n_] := Length[ Select[ Log[2, Flatten[ Partitions[n]]], IntegerQ[ # ] && # > 0 & ]]; Table[ f[n], {n, 1, 45}]

a[n_] := Sum[IntegerExponent[k, 2]*PartitionsP[n-k], {k, 1, n}]; Table[a[n], {n, 1, 50}] (* Jean-Fran├žois Alcover, Jan 28 2014 *)

CROSSREFS

Cf. A024786, A000070.

Sequence in context: A322610 A322468 A116930 * A002257 A101528 A119040

Adjacent sequences:  A073116 A073117 A073118 * A073120 A073121 A073122

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Aug 24 2002

EXTENSIONS

Edited and extended by Robert G. Wilson v, Aug 26 2002

STATUS

approved

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Last modified February 19 20:17 EST 2019. Contains 320328 sequences. (Running on oeis4.)