A023894 -id:A023894 - cp's OEIS frontend

A284289 Number of partitions of n into prime power divisors of n (not including 1).

1, 0, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 7, 1, 2, 2, 10, 1, 7, 1, 10, 2, 2, 1, 34, 2, 2, 5, 13, 1, 21, 1, 36, 2, 2, 2, 72, 1, 2, 2, 73, 1, 28, 1, 19, 13, 2, 1, 249, 2, 10, 2, 22, 1, 50, 2, 127, 2, 2, 1, 419, 1, 2, 17, 202, 2, 42, 1, 28, 2, 43, 1, 1260, 1, 2, 13, 31, 2, 49, 1, 801, 23, 2, 1, 774, 2, 2, 2, 280, 1, 608

Offset: 0

Ilya Gutkovskiy, Mar 24 2017

nonn

			a(8) = 4 because 8 has 4 divisors {1, 2, 4, 8} among which 3 are prime powers {2, 4, 8} therefore we have [8], [4, 4], [4, 2, 2] and [2, 2, 2, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for related partition-counting sequences

Cf. A014652, A018818, A023894, A066882, A225244, A211110, A246655.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> nops(ifactors(x)[2])=1, divisors(n))[]]),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 30 2017

Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[PrimePowerQ[d[[k]]]] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 90}] (* or *)
a[0]=1; a[1]=0; a[n_] := Length@IntegerPartitions[n, All, Join @@ (#[[1]]^Range[#[[2]]] & /@ FactorInteger[n])]; a /@ Range[0, 90] (* Giovanni Resta, Mar 25 2017 *)

a(n) = [x^n] Product_{p^k|n, p prime, k >= 1} 1/(1 - x^(p^k)).
a(n) = 1 if n is a prime.
a(n) = 2 if n is a semiprime.

A321347 Number of strict integer partitions of n containing no prime powers (including 1).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 4, 4, 2, 3, 4, 4, 5, 6, 5, 6, 7, 7, 9, 10, 10, 13, 12, 11, 15, 17, 16, 19, 20, 20, 25, 28, 26, 30, 33, 35, 41, 43, 42, 50, 55, 57, 64, 67, 67, 79, 86, 87, 97, 105, 109, 124, 131, 135, 151, 163, 169

Offset: 0

Gus Wiseman, Dec 11 2018

nonn

First differs from A286221 at a(30) = 6, A286221(30) = 5.

			The a(36) = 13 strict integer partitions:
  (36),
  (21,15), (22,14), (24,12), (26,10), (30,6), (35,1),
  (14,12,10), (18,12,6), (20,10,6), (20,15,1), (21,14,1),
  (15,14,6,1).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

Cf. A000607, A000961, A001597, A002095, A023893, A023894, A096258, A246655, A320322, A321346, A321378, A321665, A322452, A322454.

nn=100;
ser=Product[If[PrimePowerQ[n],1,1+x^n],{n,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

A321665 Number of strict integer partitions of n containing no 1's or prime powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 2, 2, 0, 3, 1, 3, 2, 4, 1, 5, 2, 5, 4, 6, 4, 9, 3, 8, 7, 10, 6, 13, 7, 13, 12, 16, 10, 20, 13, 22, 19, 24, 18, 32, 23, 34, 30, 37, 30, 49, 37, 50, 47, 58, 51, 73, 58, 77, 74, 89, 80, 108, 91, 116

Offset: 0

Gus Wiseman, Dec 11 2018

nonn

			The a(36) = 9 strict integer partitions:
  (36)
  (30,6)
  (21,15)
  (22,14)
  (24,12)
  (26,10)
  (18,12,6)
  (20,10,6)
  (14,12,10)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..500 from Fausto A. C. Cariboni)

Cf. A000607, A000961, A001597, A002095, A023893, A023894, A096258, A246655, A321346, A321347, A321378, A322452, A322454.

nn=100;
ser=Product[If[PrimePowerQ[n],1,1+x^n],{n,2,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

G.f.: Product_{k>=2, k not a prime power} 1 + x^k. - Joerg Arndt, Dec 22 2020

A322454 Number of multiset partitions with no constant parts of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 4, 0, 4, 0, 3, 3, 1, 0, 7, 4, 1, 9, 4, 0, 7, 0, 11, 3, 1, 5, 15, 0, 1, 4, 11

Offset: 1

Gus Wiseman, Dec 09 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(30) = 7 multiset partitions:
    {{1,1,1,2,2,3}}
   {{1,2},{1,1,2,3}}
   {{1,3},{1,1,2,2}}
   {{2,3},{1,1,1,2}}
   {{1,1,2},{1,2,3}}
   {{1,1,3},{1,2,2}}
  {{1,2},{1,2},{1,3}}

Cf. A000688, A000961, A001055, A001597, A023893, A023894, A181821, A318284, A320322, A321407, A321760, A322260, A322452.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[mps[nrmptn[n]],Min@@Length/@Union/@#>1&]],{n,20}]

A356066 Numbers with a prime index that is not a prime-power. Complement of A355743.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 52, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 84, 86, 87, 88, 89, 90, 91, 92, 94, 96, 98, 100, 101

Offset: 1

Gus Wiseman, Jul 31 2022

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.

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}

The complement is A355743, counted by A023894.
The squarefree complement is A356065, counted by A054685.
Allowing prime index 1 gives A356064, complement A302492.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A034699 gives the maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A076610, A085970, A106244, A302493, A302601, A330946, A354911.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@PrimePowerQ/@primeMS[#]&]

Union of A299174 and A356064.

A280151 Expansion of Product_{k>=1} 1/(1 - floor(1/omega(2k+1))x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 23, 26, 29, 33, 37, 42, 46, 53, 58, 66, 74, 81, 91, 101, 113, 124, 139, 153, 169, 188, 207, 228, 252, 278, 304, 336, 369, 405, 444, 487, 533, 583, 640, 697, 763, 832, 908, 990, 1078, 1175, 1278

Offset: 0

Ilya Gutkovskiy, Dec 27 2016

nonn

Number of partitions of n into odd prime powers (1 excluded).

			a(12) = 3 because we have [9, 3], [7, 5], [3, 3, 3, 3].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for related partition-counting sequences

Cf. A001221, A018819, A023894, A061345, A246655, A280152.

nmax = 67; CoefficientList[Series[Product[1/(1 - Floor[1/PrimeNu[2 k + 1]] x^(2 k + 1)), {k, 1, nmax}], {x, 0, nmax}], x]

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

A307726 Number of partitions of n into 2 prime powers (not including 1).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 2, 4, 3, 4, 4, 4, 2, 4, 3, 4, 4, 4, 3, 5, 3, 6, 4, 7, 4, 7, 2, 5, 4, 6, 3, 5, 3, 5, 5, 6, 2, 7, 3, 7, 4, 6, 2, 8, 3, 7, 4, 6, 2, 7, 3, 6, 4, 7, 2, 9, 2, 7, 5, 7, 2, 9, 3, 7, 6, 7, 3, 9, 2, 8, 4, 6, 4, 10, 3, 9, 4, 7, 3, 11, 4, 8, 3, 7, 2, 10, 2, 8, 3, 8

Offset: 0

Ilya Gutkovskiy, Apr 24 2019

			a(10) = 3 because we have [8, 2], [7, 3] and [5, 5].

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

Cf. A000961, A023894, A061358, A071068, A071330, A071331, A246655, A280242, A307727.

# note that this requires A246655 to be pre-computed
f:= proc(n, k, pmax) option remember;
  local t, p, j;
  if n = 0 then return `if`(k=0, 1, 0) fi;
  if k = 0 then return 0 fi;
  if n > k*pmax then return 0 fi;
  t:= 0:
  for p in A246655 do
    if p > pmax then return t fi;
    t:= t + add(procname(n-j*p, k-j, min(p-1, n-j*p)), j=1..min(k, floor(n/p)))
  od;
  t
end proc:
map(f, [$0..100]); # Robert Israel, Apr 29 2019

Array[Count[IntegerPartitions[#, {2}], _?(AllTrue[#, PrimePowerQ] &)] &, 101, 0]

a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^A246655(k)).

A300580 Number of partitions of n into prime power parts (not including 1) that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 1, 3, 2, 11, 1, 18, 6, 9, 5, 43, 5, 65, 7, 31, 30, 137, 5, 115, 59, 84, 26, 379, 19, 519, 42, 213, 197, 323, 23, 1267, 340, 489, 50, 2213, 107, 2897, 221, 375, 938, 4871, 61, 3733, 662, 2193, 553, 10218, 409, 4241, 310, 4341, 3685, 20586, 154, 25792, 5635, 2862, 990, 12806

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

			a(10) = 2 because we have [7, 3] and [4, 3, 3].

Index entries for sequences related to partitions

Cf. A023894, A098743, A128515, A246655, A284289, A300584.

Table[SeriesCoefficient[Product[1/(1 - Boole[Mod[n, k] != 0 && PrimePowerQ[k]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 65}]

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)).

A318914 Expansion of e.g.f. Product_{p prime, k>=1} 1/(1 - x^(p^k))^(1/(p^k)).

Original entry on oeis.org

1, 0, 1, 2, 15, 44, 475, 2274, 33313, 227240, 2920041, 26754650, 469513231, 4613913732, 85842524755, 1174844041994, 24672317426625, 334246510927184, 7985602649948113, 127351500133158450, 3282809137540001551, 60776696924693716700, 1556379682561575238731, 32568139442090869594802

Offset: 0

Ilya Gutkovskiy, Sep 05 2018

nonn

Cf. A001222, A023894, A028342, A206303, A318912, A318913.

seq(n!*coeff(series(exp(add(bigomega(k)*x^k/k,k=1..100)),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 09 2019

nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(Boole[PrimePowerQ[k]]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[PrimeOmega[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[PrimeOmega[k] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

E.g.f.: exp(Sum_{k>=1} bigomega(k)*x^k/k), where bigomega(k) = number of prime divisors of k counted with multiplicity (A001222).

cp's OEIS Frontend

A284289 Number of partitions of n into prime power divisors of n (not including 1).

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A321347 Number of strict integer partitions of n containing no prime powers (including 1).

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A321665 Number of strict integer partitions of n containing no 1's or prime powers.

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A322454 Number of multiset partitions with no constant parts of a multiset whose multiplicities are the prime indices of n.

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A356066 Numbers with a prime index that is not a prime-power. Complement of A355743.

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A280151 Expansion of Product_{k>=1} 1/(1 - floor(1/omega(2*k+1))*x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).

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A307726 Number of partitions of n into 2 prime powers (not including 1).

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A300580 Number of partitions of n into prime power parts (not including 1) that do not divide n.

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A280151 Expansion of Product_{k>=1} 1/(1 - floor(1/omega(2k+1))x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).