A064574 -id:A064574 - cp's OEIS frontend

A064577 First differences of A064574.

0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 4, 0, 3, 0, 4, 1, 1, 0, 6, 1, 1, 2, 4, 0, 4, 0, 6, 1, 1, 1, 9, 0, 1, 1, 7, 0, 5, 0, 5, 3, 1, 0, 11, 1, 3, 1, 6, 0, 6, 1, 8, 1, 1, 0, 12, 0, 1, 3, 11, 1, 5, 0, 8, 1, 4, 0, 17, 0, 1, 3, 8, 1, 6, 0, 15, 4, 1, 0, 17, 1, 1, 1, 13, 0, 11, 1, 10, 1, 1, 1, 21, 0, 3, 4, 16

Offset: 1

Marc LeBrun, Sep 20 2001

Apparently a(n)=0 when n+1 is prime. Diverges from A028422(n+1) at n=19.

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A064572, A064573, A064574, A064575, A064576, A028422.

a(n) = A064574(n+1) - A064574(n). - Antti Karttunen, Feb 24 2020

A064573 Number of partitions of n into parts which are all powers of the same prime.

Original entry on oeis.org

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

Marc LeBrun, Sep 20 2001

The exponents cannot all be zero.

			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)

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

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

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

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

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

Name clarified by Andrew Howroyd, Dec 29 2017

A064572 Number of ways to partition n into parts which are all powers of some integer k.

Original entry on oeis.org

0, 1, 2, 5, 6, 10, 11, 17, 20, 26, 27, 38, 39, 47, 51, 65, 66, 82, 83, 102, 107, 123, 124, 153, 156, 178, 185, 216, 217, 254, 255, 297, 304, 342, 346, 408, 409, 457, 466, 535, 536, 609, 610, 690, 704, 780, 781, 895, 898, 998, 1009, 1130, 1131, 1263, 1268, 1418

Offset: 1

Marc LeBrun, Sep 20 2001

Number of ways to partition n as Sum_i k^e_i, where the exponents e_i are not all 0.

The exponents cannot all be 0, e.g. a(2)=1 arises from 2^1, and does not include 2^0+2^0. - Shujing Lyu, Apr 23 2016

			a(4)=5: 4^1, 3^1+3^0, 2^2, 2*2^1, 2^1+2*2^0.

Andrew Howroyd, Table of n, a(n) for n = 1..1000 (terms 1..220 from Shujing Lyu)

Cf. A028422, A064573, A064574, A064575, A064576, A064577, A292477.

first(n)={Vec(sum(k=2, n, 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

G.f.: Sum_{k>=2} 1/(Product_{r>=0} 1-x^(k^r)) - 1/(1-x). - Andrew Howroyd, Dec 29 2017

A064575 First differences of A064572, where A064572(n) is the number of ways to partition n into parts which are all powers of some integer.

Original entry on oeis.org

1, 1, 3, 1, 4, 1, 6, 3, 6, 1, 11, 1, 8, 4, 14, 1, 16, 1, 19, 5, 16, 1, 29, 3, 22, 7, 31, 1, 37, 1, 42, 7, 38, 4, 62, 1, 48, 9, 69, 1, 73, 1, 80, 14, 76, 1, 114, 3, 100, 11, 121, 1, 132, 5, 150, 14, 142, 1, 193, 1, 168, 20, 213, 5, 223, 1, 247, 17, 247, 1, 319, 1, 286, 25, 339, 4, 355

Offset: 1

Marc LeBrun, Sep 20 2001

Apparently a(n)=1 when n+1 is prime.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A064572, A064573, A064574, A064576, A064577, A028422.

up_to = 200;
A064572list(n) = {Vec(sum(k=2, n, 1/prod(r=0, logint(n, k), 1-x^(k^r) + O(x*x^n)) - 1/(1-x), 0), -n)}; \\ From A064572 by Andrew Howroyd, Dec 29 2017
v064572 = A064572list(1+up_to);
A064572(n) = v064572[n];
A064575(n) = (A064572(1+n)-A064572(n)); \\ Antti Karttunen, Jan 24 2025

A064576 First differences of A064573, where A064573(n) is the number of partitions of n into parts which are all powers of the same prime.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 8, 1, 7, 3, 10, 1, 13, 1, 15, 4, 15, 1, 23, 2, 21, 5, 27, 1, 33, 1, 36, 6, 37, 3, 53, 1, 47, 8, 62, 1, 68, 1, 75, 11, 75, 1, 103, 2, 97, 10, 115, 1, 126, 4, 142, 13, 141, 1, 181, 1, 167, 17, 202, 4, 218, 1, 239, 16, 243, 1, 302, 1, 285, 22, 331, 3, 349

Offset: 1

Marc LeBrun, Sep 20 2001

Apparently a(n)=1 when n+1 is prime.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A064572, A064573, A064574, A064575, A064577, A028422.

up_to = 200;
A064573list(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)}; \\ From A064573 by Andrew Howroyd, Dec 29 2017
v064573 = A064573list(1+up_to);
A064573(n) = v064573[n];
A064576(n) = (A064573(1+n)-A064573(n)); \\ Antti Karttunen, Jan 24 2025

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A064577 First differences of A064574.

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A064573 Number of partitions of n into parts which are all powers of the same prime.

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A064572 Number of ways to partition n into parts which are all powers of some integer k.

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A064575 First differences of A064572, where A064572(n) is the number of ways to partition n into parts which are all powers of some integer.

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A064576 First differences of A064573, where A064573(n) is the number of partitions of n into parts which are all powers of the same prime.

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