A023895 Number of partitions of n into composite parts.
1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 4, 1, 4, 2, 7, 2, 9, 3, 12, 6, 15, 6, 23, 11, 26, 15, 37, 19, 48, 26, 61, 39, 78, 47, 105, 65, 126, 88, 167, 111, 211, 146, 264, 196, 331, 241, 426, 318, 519, 408, 657, 511, 820, 651, 1010, 833, 1252, 1028, 1564, 1301, 1900
Offset: 0
Keywords
Examples
a(12) = 4 because 12 = 4 + 4 + 4 = 6 + 6 = 4 + 8 = 12 (itself a composite number).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..5000 (terms n = 0..150 from Reinhard Zumkeller)
Programs
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Haskell
a023895 = p a002808_list where p _ 0 = 1 p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m -- Reinhard Zumkeller, Jan 15 2012
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Maple
g:=(1-x)*product((1-x^ithprime(j))/(1-x^j),j=1..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..62); # Emeric Deutsch, Apr 03 2006 # second Maple program: b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)+ `if`(i>n or isprime(i), 0, b(n-i, i)))) end: a:= n-> b(n$2): seq(a(n), n=0..70); # Alois P. Heinz, May 29 2013
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Mathematica
Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; CoefficientList[ Series[1/Product[1 - x^Composite[i], {i, 1, 50}], {x, 0, 75}], x] (* Second program: *) b[n_, i_] := b[n, i] = If[n==0, 1, If[i<2, 0, b[n, i-1] + If[i>n || PrimeQ[i], 0, b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)
Formula
G.f.: (1-x)*Product_{j>=1} (1-x^prime(j))/(1-x^j). - Emeric Deutsch, Apr 03 2006
Extensions
More terms from Reinhard Zumkeller, Aug 22 2007
Comments