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 A064573 Number of partitions of n into parts which are all powers of the same prime. 30
 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 (list; graph; refs; listen; history; text; internal format)
 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 * A065300 A080403 A280050 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

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Last modified May 26 02:43 EDT 2020. Contains 334613 sequences. (Running on oeis4.)