A073335 Total number of prime power parts in all partitions of n.
0, 1, 2, 5, 8, 15, 23, 39, 58, 89, 128, 189, 264, 375, 515, 713, 960, 1301, 1726, 2298, 3011, 3948, 5113, 6625, 8492, 10880, 13825, 17545, 22108, 27823, 34800, 43465, 54003, 66983, 82709, 101960, 125180, 153432, 187397, 228490, 277707, 336972
Offset: 1
Examples
a(4)=5 because in all partitions of 4 we have 5 powers of primes (shown between parentheses): (4), (3)1, (2)(2), (2)11, 1111.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
Programs
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Maple
with(numtheory): with(combinat): a:= n-> add(bigomega(k)*numbpart(n-k), k=1..n): seq(a(n), n=1..46); # Emeric Deutsch, Feb 26 2005
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Mathematica
Table[Sum[PrimeOmega[k]*PartitionsP[n - k], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, May 05 2017 *) -
PARI
a(n) = sum(k=1, n, bigomega(k)*numbpart(n-k)); \\ Michel Marcus, May 05 2017
Formula
a(n) = Sum_{k=1..n} bigomega(k)*numbpart(n-k).
G.f.: Sum_{i>=2} floor(1/omega(i))*x^i/(1 - x^i) / Product_{j>=1} (1 - x^j), where omega() is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 24 2017
Extensions
More terms from Emeric Deutsch, Feb 26 2005