A064573 Number of partitions of n into parts which are all powers of the same prime.
0, 1, 2, 4, 5, 8, 9, 13, 15, 20, 21, 29, 30, 37, 40, 50, 51, 64, 65, 80, 84, 99, 100, 123, 125, 146, 151, 178, 179, 212, 213, 249, 255, 292, 295, 348, 349, 396, 404, 466, 467, 535, 536, 611, 622, 697, 698, 801, 803, 900, 910, 1025, 1026, 1152, 1156, 1298, 1311
Offset: 1
Examples
a(5)=5: 5^1, 3^1+2*3^0, 2^2+1, 2*2^1+1, 2^1+3*2^0
From _Gus Wiseman_, Oct 10 2018: (Start)
The a(2) = 1 through a(9) = 15 integer partitions:
(2) (3) (4) (5) (33) (7) (8) (9)
(21) (22) (41) (42) (331) (44) (81)
(31) (221) (51) (421) (71) (333)
(211) (311) (222) (511) (422) (441)
(2111) (411) (2221) (2222) (711)
(2211) (4111) (3311) (4221)
(3111) (22111) (4211) (22221)
(21111) (31111) (5111) (33111)
(211111) (22211) (42111)
(41111) (51111)
(221111) (222111)
(311111) (411111)
(2111111) (2211111)
(3111111)
(21111111)
(End)
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1000
Programs
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Mathematica
Table[Length[Select[IntegerPartitions[n],PrimePowerQ[Times@@#]&]],{n,30}] (* Gus Wiseman, Oct 10 2018 *) -
PARI
first(n)={Vec(sum(k=2, n, if(isprime(k), 1/prod(r=0, logint(n,k), 1-x^(k^r) + O(x*x^n)) - 1/(1-x), 0)), -n)} \\ Andrew Howroyd, Dec 29 2017
Formula
G.f.: Sum_{k>=1} 1/(Product_{r>=0} 1-x^(prime(k)^r)) - 1/(1-x). - Andrew Howroyd, Dec 29 2017
Extensions
Name clarified by Andrew Howroyd, Dec 29 2017
Comments