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

1,3

Comments

The exponents cannot all be zero.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{k>=1} 1/(Product_{r>=0} 1-x^(prime(k)^r)) - 1/(1-x). - Andrew Howroyd, Dec 29 2017

Example

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)

Mathematica

Table[Length[Select[IntegerPartitions[n], PrimePowerQ[Times@@#]&]], {n, 30}] (* Gus Wiseman, Oct 10 2018 *)

Prog

(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

Crossrefs

Cf. A028422, A064572, A064574, A064575, A064576, A064577, A319071, A320322, A320325.

Sequence in context: A102821 A101881 A143989 * A372953 A065300 A080403

Adjacent sequences: A064570 A064571 A064572 * A064574 A064575 A064576

Keyword

easy,nonn

Author

Marc LeBrun, Sep 20 2001

Extensions

Name clarified by Andrew Howroyd, Dec 29 2017

Status

approved