A064573 -id:A064573 - cp's OEIS frontend

A322526 Number of integer partitions of n whose product of parts is a squarefree number.

1, 1, 2, 3, 3, 5, 6, 8, 9, 10, 13, 15, 17, 21, 24, 27, 30, 36, 41, 46, 51, 57, 65, 73, 82, 90, 101, 109, 121, 134, 150, 164, 177, 193, 214, 232, 253, 278, 300, 324, 351, 386, 419, 452, 484, 521, 563, 610, 658, 706, 758, 809, 868, 938, 1006, 1071, 1140, 1220, 1307

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

The parts of such a partition must also be squarefree and distinct except for any number of 1's.

			The a(8) = 9 partitions are (53), (71), (521), (611), (5111), (32111), (311111), (2111111), (11111111). Missing from this list are (8), (62), (44), (431), (422), (4211), (41111), (332), (3311), (3221), (2222), (22211), (221111).

Partial sums of A322530.

Cf. A001055, A003963, A005117, A064573 , A073576, A096276, A301987, A302505, A319005, A319916, A320322, A322527, A322529, A322531.

Table[Length[Select[IntegerPartitions[n],SquareFreeQ[Times@@#]&]],{n,30}]

A319071 Number of integer partitions of n whose product of parts is a perfect power and whose parts all have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 2, 0, 3, 2, 3, 0, 4, 1, 4, 3, 7, 1, 7, 1, 8, 6, 8, 0, 15, 5, 12, 6, 15, 4, 22, 4, 24, 12, 22, 8, 35, 7, 30, 16, 42, 9, 50, 9, 50, 30, 53, 7, 79, 22, 72, 33, 87, 21, 109, 26, 111, 55, 117, 24, 168, 40, 149, 65, 178, 59

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

The positions of zeros appear to be A048278.

			The a(4) = 2 through a(16) = 7 integer partitions (G = 16):
  4   33   8     9    55     66      94  77       555     G
  22  222  44    333  3322   444         5522     33333   88
           2222       22222  3333        332222   333222  664
                             222222      2222222          4444
                                                          5533
                                                          333322
                                                          22222222

Cf. A003963, A048278, A064573, A279787, A305551, A319056, A319066, A319169, A320322, A320323.

Table[Length[Select[IntegerPartitions[n],And[GCD@@FactorInteger[Times@@#][[All,2]]>1,SameQ@@PrimeOmega/@#]&]],{n,30}]

A322527 Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 13, 18, 21, 31, 34, 45, 51, 63, 72, 88, 97, 120, 128, 158, 174, 201, 222, 264, 287, 333, 359, 416, 441, 518, 557, 631, 684, 770, 833, 954, 1017, 1141, 1222, 1378, 1475, 1643, 1755, 1939, 2097, 2327, 2471, 2758, 2928, 3233, 3470, 3813, 4085

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

			The a(1) = 1 through a(8) = 18 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (52)       (44)
             (111)  (31)    (41)     (42)      (61)       (53)
                    (211)   (221)    (51)      (331)      (71)
                    (1111)  (311)    (222)     (421)      (422)
                            (2111)   (321)     (511)      (521)
                            (11111)  (411)     (2221)     (611)
                                     (2211)    (3211)     (2222)
                                     (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
Missing from the list for n = 7 through 9:
  (43)   (62)    (54)
  (322)  (332)   (63)
         (431)   (432)
         (3221)  (522)
                 (621)
                 (3222)
                 (3321)
                 (4311)
                 (32211)

Cf. A003963, A005117, A038041, A062503, A064573, A072774, A295193, A302505, A306021, A319169, A320322, A322526, A322528, A322530.

Table[Length[Select[IntegerPartitions[n],SameQ@@Last/@FactorInteger[Times@@#]&]],{n,30}]

A322530 Number of integer partitions of n with no 1's whose product of parts is a squarefree number.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 1, 1, 3, 2, 2, 4, 3, 3, 3, 6, 5, 5, 5, 6, 8, 8, 9, 8, 11, 8, 12, 13, 16, 14, 13, 16, 21, 18, 21, 25, 22, 24, 27, 35, 33, 33, 32, 37, 42, 47, 48, 48, 52, 51, 59, 70, 68, 65, 69, 80, 87, 90, 103, 100, 96, 103, 123, 128, 135, 136, 132, 153

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

Such a partition must be strict and its parts must also be squarefree.

			The a(26) = 11 integer partitions:
  (26),
  (15,11), (19,7), (21,5), (23,3),
  (13,7,6), (13,10,3), (13,11,2), (17,7,2), (19,5,2),
  (11,7,5,3).

First differences of A322526.

Cf. A002865, A003963, A005117, A064573, A072774, A073576, A302505, A319005, A319057, A319169, A320322, A322527, A322528, A322529.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&SquareFreeQ[Times@@#]&]],{n,30}]

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

A064574 Number of partitions of n into parts which are all powers of the same composite.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 4, 5, 6, 6, 9, 9, 10, 11, 15, 15, 18, 18, 22, 23, 24, 24, 30, 31, 32, 34, 38, 38, 42, 42, 48, 49, 50, 51, 60, 60, 61, 62, 69, 69, 74, 74, 79, 82, 83, 83, 94, 95, 98, 99, 105, 105, 111, 112, 120, 121, 122, 122, 134, 134, 135, 138, 149, 150, 155, 155

Offset: 1

Marc LeBrun, Sep 20 2001

The exponents cannot all be 0.

Diverges from A028422 at n=20.

			a(8)=4: 8^1, 6^1+2*6^0, 2*4^1, 4^1+4*2^0

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

A002808, A064572, A064573, A064575, A064576, A064577, A028422.

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^(A002808(k)^r)) - 1/(1-x). - Andrew Howroyd, Dec 29 2017

Name clarified by 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

A064577 First differences of A064574.

Original entry on oeis.org

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

A320698 Numbers whose product of prime indices is a prime power (A246655).

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 31, 34, 36, 38, 40, 41, 42, 44, 46, 48, 49, 50, 53, 54, 56, 57, 59, 62, 63, 67, 68, 72, 76, 80, 81, 82, 83, 84, 88, 92, 96, 97, 98, 100, 103, 106, 108, 109, 112, 114, 115, 118, 121, 124

Offset: 1

Gus Wiseman, Oct 19 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also numbers whose prime indices are all powers of a common prime number.

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (2), (3), (1,2), (4), (2,2), (1,3), (5), (1,1,2), (1,4), (7), (1,2,2), (8), (1,1,3), (2,4), (1,5), (9), (1,1,1,2), (3,3), (2,2,2), (1,1,4), (11), (1,7), (1,1,2,2), (1,8), (1,1,1,3), (13), (1,2,4), (1,1,5), (1,9), (1,1,1,1,2), (4,4), (1,3,3), (16), (1,2,2,2), (1,1,1,4), (2,8).

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A000961, A001597, A003963, A047968, A056239, A064573, A112798, A246547, A246655, A279787, A320322, A320325, A320699, A320700.

Select[Range[100],PrimePowerQ[Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]]&]

is(n) = my(f=factor(n)[, 1]~, p=1); for(k=1, #f, p=p*primepi(f[k])); isprimepower(p) \\ Felix Fröhlich, Oct 20 2018

A322528 Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 5, 4, 7, 2, 7, 4, 7, 7, 9, 3, 10, 5, 12, 9, 8, 6, 14, 10, 12, 10, 14, 11, 20, 13, 18, 13, 16, 16, 25, 16, 19, 20, 26, 18, 30, 19, 27, 26, 27, 22, 38, 30, 37, 28, 38, 32, 43, 37, 46, 40, 47, 40, 66, 49, 58, 56, 64, 56, 73, 58, 76, 70, 85

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

			The a(1) = 1 through a(8) = 5 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (52)       (44)
                    (1111)  (11111)  (222)     (1111111)  (53)
                                     (111111)             (2222)
                                                          (11111111)

Cf. A002865, A003963, A005117, A064573 , A072774, A295193, A302505, A306017, A319169, A320322, A322526, A322527, A322529, A322530, A322531.

Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SameQ@@Last/@FactorInteger[Times@@#]]&]],{n,30}]

More terms from Alois P. Heinz, Dec 14 2018

cp's OEIS Frontend

A322526 Number of integer partitions of n whose product of parts is a squarefree number.

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A319071 Number of integer partitions of n whose product of parts is a perfect power and whose parts all have the same number of prime factors, counted with multiplicity.

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A322527 Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).

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A322530 Number of integer partitions of n with no 1's whose product of parts is a squarefree number.

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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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A064574 Number of partitions of n into parts which are all powers of the same composite.

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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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A064577 First differences of A064574.

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A320698 Numbers whose product of prime indices is a prime power (A246655).

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