A023893 -id:A023893 - cp's OEIS frontend

A137792 Number of partitions of n into parts with no prime gaps in their factorization.

1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 98, 129, 168, 218, 279, 358, 453, 573, 719, 899, 1118, 1389, 1712, 2106, 2581, 3152, 3835, 4657, 5632, 6797, 8177, 9813, 11746, 14033, 16715, 19873, 23575, 27910, 32972, 38886, 45765, 53770, 63062, 73833

Offset: 1

Reinhard Zumkeller, Feb 11 2008

nonn

			a(12) = A000041(12) - #{10+2, 10+1+1} = 77 - 2 = 75.

Cf. A073491, A000009, A000607, A023893, A137793, A137791.

A322912 Number of integer partitions of n whose parts are all powers of the same squarefree number.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 17, 23, 24, 33, 34, 42, 46, 56, 57, 71, 72, 88, 93, 109, 110, 134, 136, 158, 163, 191, 192, 229, 230, 266, 273, 311, 315, 370, 371, 419, 428, 491, 492, 565, 566, 642, 654, 730, 731, 836, 838, 936

Offset: 0

Gus Wiseman, Dec 30 2018

nonn

First differs from A072720 at a(12) = 33, A072720(12) = 34.

			The a(1) = 1 through a(8) = 15 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (42)      (331)      (71)
                    (211)   (311)    (51)      (421)      (422)
                    (1111)  (2111)   (222)     (511)      (611)
                            (11111)  (411)     (2221)     (2222)
                                     (2211)    (4111)     (3311)
                                     (3111)    (22111)    (4211)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Cf. A000961, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A302593, A322847, A322900, A322901, A322911.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
Table[Length[Select[IntegerPartitions[n],And[And@@powsqfQ/@#,SameQ@@radbase/@DeleteCases[#,1]]&]],{n,30}]

A322968 Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 1, 4, 2, 6, 1, 9, 1, 8, 4, 10, 1, 14, 1, 16, 5, 16, 1, 24, 2, 22, 5, 28, 1, 37, 1, 36, 7, 38, 4, 55, 1, 48, 9, 63, 1, 73, 1, 76, 12, 76, 1, 105, 2, 98, 11, 116, 1, 128, 5, 143, 14, 142, 1, 186, 1, 168, 18, 202, 5, 223, 1, 240, 17, 247, 1, 305, 1, 286, 23

Offset: 0

Gus Wiseman, Jan 01 2019

nonn

First differs from A072721 at a(12) = 9, A072721(12) = 10.

First differs from A379957 at a(16) = 10, A379957(16) = 9.

			The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)  (66)
            (22)       (33)        (44)    (333)  (55)          (84)
                       (42)        (422)          (82)          (93)
                       (222)       (2222)         (442)         (444)
                                                  (4222)        (822)
                                                  (22222)       (3333)
                                                                (4422)
                                                                (42222)
                                                                (222222)
The a(20) = 16 integer partitions:
  (10,10), (16,4),
  (8,8,4), (16,2,2),
  (5,5,5,5), (8,4,4,4), (8,8,2,2),
  (4,4,4,4,4), (8,4,4,2,2),
  (4,4,4,4,2,2), (8,4,2,2,2,2),
  (4,4,4,2,2,2,2), (8,2,2,2,2,2,2),
  (4,4,2,2,2,2,2,2),
  (4,2,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2,2).

Andrew Howroyd, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions.

Cf. A001597, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A102430, A322900, A322903, A322911, A322912, A379957.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],And@@powsqfQ/@#,SameQ@@radbase/@#]&]],{n,30}]

a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&issquarefree(d), polcoef(1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)), n))))} \\ Andrew Howroyd, Jan 23 2025

seq(n)={Vec(1 + sum(d=2, n, if(issquarefree(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ Andrew Howroyd, Jan 23 2025

From  Andrew Howroyd, Jan 23 2025: (Start)
G.f.: 1 + Sum_{k>=2} -1 + 1/Product_{j>=1} (1 - x^(A005117(k)^j)).
a(p) = 1 for prime p. (End)

a(66) onwards from Andrew Howroyd, Jan 23 2025

A280125 Expansion of Product_{k>=1} 1/((1 - x^(prime(k)^2))*(1 - x^(prime(k)^3))).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 3, 2, 1, 0, 3, 2, 1, 0, 4, 4, 2, 2, 4, 4, 2, 2, 5, 6, 4, 4, 7, 6, 4, 4, 8, 8, 6, 7, 10, 10, 6, 7, 11, 13, 9, 10, 15, 15, 12, 10, 16, 18, 16, 14, 20, 22, 19, 17, 21, 25, 23, 22, 26, 29, 28, 25, 30, 32, 33, 31, 37, 38, 38, 37

Offset: 0

Ilya Gutkovskiy, Dec 26 2016

nonn

Number of partitions of n into parts that are squares of primes (A001248) or cubes of primes (A030078).

			a(16) = 3 because we have [8, 8], [8, 4, 4] and [4, 4, 4, 4].

Index entries for related partition-counting sequences

Cf. A000607, A001248, A023893, A030078, A090677, A111901, A131799, A168363, A279760, A280126.

nmax = 100; CoefficientList[Series[Product[1/((1 - x^Prime[k]^2) (1 - x^Prime[k]^3)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/((1 - x^(prime(k)^2))*(1 - x^(prime(k)^3))).

A280586 Expansion of Product_{p prime, k>=2} 1/(1 - x^(p^k)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 4, 2, 1, 0, 4, 2, 1, 0, 6, 5, 2, 2, 6, 5, 2, 2, 10, 8, 5, 4, 12, 8, 5, 4, 16, 14, 8, 9, 18, 16, 8, 9, 24, 23, 15, 14, 30, 25, 18, 14, 36, 36, 26, 25, 42, 42, 29, 28, 52, 54, 42, 40, 65, 60, 50, 43, 78, 78, 65, 63, 93, 92, 73, 72, 110, 117, 96, 94, 135, 133, 114, 103, 158, 166, 145

Offset: 0

Ilya Gutkovskiy, Jan 06 2017

nonn

Number of partitions of n into proper prime powers (A246547).

			a(16) = 4 because we have [16], [8, 8], [8, 4, 4] and [4, 4, 4, 4].

Eric Weisstein's World of Mathematics, Prime Power
Index entries for related partition-counting sequences

Cf. A023893, A023894, A246547.

nmax = 90; CoefficientList[Series[Product[1/(1 - Sign[PrimeOmega[k] - 1] Floor[1/PrimeNu[k]] x^k), {k, 2, nmax}], {x, 0, nmax}], x]

x='x+O('x^68); Vec(prod(k=2, 67, 1/(1 - sign(bigomega(k) - 1) * (1\omega(k)) * x^k))) \\ Indranil Ghosh, Apr 03 2017

G.f.: Product_{p prime, k>=2} 1/(1 - x^(p^k)).

A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23

Offset: 1

Gus Wiseman, Dec 14 2018

			24 does not belong to the sequence because there are integer partitions of 24 containing no 1's or prime powers, namely: (24), (18,6), (14,10), (12,12), (12,6,6), (6,6,6,6).

Cf. A000607, A002095, A023893, A023894, A064573, A078135, A101417, A246655, A320322, A322452, A322454, A322547.

nn=100;
ser=Product[If[n==1||PrimePowerQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A322547 Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 71, 79

Offset: 1

Gus Wiseman, Dec 14 2018

			48 does not belong to the sequence because there are integer partitions of 48 containing no 1's, squarefree numbers, or prime powers, namely: (48), (36,12), (28,20), (24,24), (24,12,12), (18,18,12), (12,12,12,12).

Cf. A000607, A002095, A005117, A023893, A023894, A064573, A078135, A101417, A246655, A322546.

nn=100;
ser=Product[If[PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A322911 Numbers whose prime indices are all powers of the same squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 62, 63, 64, 67, 68, 72, 73, 76, 79, 80, 81, 82, 83, 84, 86, 88, 92

Offset: 1

Gus Wiseman, Dec 30 2018

nonn

The complement is {15, 30, 33, 35, 37, 39, 45, ...}. First differs from A318991 at a(33) = 38, A318991(33) = 37.

A multiset multisystem is a finite multiset of finite multisets. 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. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The sequence lists all MM-numbers of multiset multisystems whose dual is constant, i.e. of the form {x,x,x,...,x} for some multiset x.

			The prime indices of 756 are {1,1,2,2,2,4}, which are all powers of 2, so 756 belongs to the sequence.
The prime indices of 841 are {10,10}, which are all powers of 10, so 841 belongs to the sequence.
The prime indices of 2645 are {3,9,9}, which are all powers of 3, so 2645 belongs to the sequence.
The prime indices of 3178 are {1,4,49}, which are all powers of squarefree numbers but not of the same squarefree number, so 3178 does not belong to the sequence.
The prime indices of 30599 are {12,144}, which are all powers of the same number 12, but this number is not squarefree, so 30599 does not belong to the sequence.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (3,2), (3,2,1), (5,2), (4,3), (6,2), (3,2,2), (7,2), (5,3), (3,2,1,1), (6,3), (5,2,1), (9,2), (4,3,1), (3,3,2), (5,4), (6,2,1), (7,3), (10,2), (3,2,2,1), (6,4), (11,2), (8,3), (5,2,2).

Cf. A000688, A000961, A001597, A005117, A023893, A052410, A056239, A072720, A072774, A302242, A302593, A318400, A322847, A322901, A322912.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
sqfker[n_]:=Times@@First/@FactorInteger[n];
Select[Range[100],And[And@@powsqfQ/@primeMS[#],SameQ@@sqfker/@DeleteCases[primeMS[#],1]]&]

A331925 Number of compositions (ordered partitions) of n into distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 10, 11, 17, 19, 48, 49, 62, 85, 120, 258, 175, 337, 464, 631, 646, 932, 1686, 1991, 2122, 2455, 4118, 4545, 6010, 6481, 13302, 14383, 16177, 16912, 26454, 32024, 35468, 42389, 57334, 107708, 73830, 125629, 142560, 200377, 172752, 244624

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

nonn

			a(6) = 10 because we have [5, 1], [4, 2], [3, 2, 1], [3, 1, 2], [2, 4], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].

Robert Israel, Table of n, a(n) for n = 0..250
Index entries for sequences related to compositions

Cf. A000961, A023893, A106244, A219107, A280543, A331847.

N:= 50: # for a(0)..a(N)
P:= select(isprime, [2,seq(i,i=3..N,2)]):
PP:= sort([1,seq(seq(p^j, j = 1 .. ilog[p](N)),p=P)]):G:= 1:
for s in PP do
  G:= G + series(G*x*y^s,y,N+1);
od:
G:= convert(G,polynom):
T:= add(coeff(G,x,i)*i!,i=0..N):
seq(coeff(T,y,i),i=0..N); # Robert Israel, Jun 28 2024

A339241 Number of partitions of n into prime power parts (including 1) where every part appears at least 2 times.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 2, 6, 5, 9, 7, 15, 11, 21, 19, 31, 27, 46, 40, 63, 60, 88, 83, 124, 117, 166, 165, 224, 222, 303, 301, 399, 407, 525, 537, 691, 707, 893, 929, 1153, 1202, 1485, 1550, 1890, 1992, 2400, 2534, 3040, 3212, 3818, 4059, 4781, 5089, 5972, 6359, 7412

Offset: 0

Ilya Gutkovskiy, Nov 28 2020

nonn

			a(6) = 4 because we have [3, 3], [2, 2, 2], [2, 2, 1, 1] and [1, 1, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000961, A007690, A023893, A161077, A339218, A339242.

nmax = 56; CoefficientList[Series[(1 + x^2/(1 - x)) Product[1 + Boole[PrimePowerQ[k]] x^(2 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: (1 + x^2 / (1 - x)) * Product_{p prime, k>=1} (1 + x^(2*p^k) / (1 - x^(p^k))).

cp's OEIS Frontend

A137792 Number of partitions of n into parts with no prime gaps in their factorization.

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A322912 Number of integer partitions of n whose parts are all powers of the same squarefree number.

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A322968 Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.

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A280125 Expansion of Product_{k>=1} 1/((1 - x^(prime(k)^2))*(1 - x^(prime(k)^3))).

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A280586 Expansion of Product_{p prime, k>=2} 1/(1 - x^(p^k)).

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A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

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A322547 Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.

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A322911 Numbers whose prime indices are all powers of the same squarefree number.

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A331925 Number of compositions (ordered partitions) of n into distinct prime powers (including 1).

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