A023893 -id:A023893 - cp's OEIS frontend

A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

13, 26, 29, 37, 39, 43, 47, 52, 58, 61, 65, 71, 73, 74, 78, 79, 86, 87, 89, 91, 94, 101, 104, 107, 111, 113, 116, 117, 122, 129, 130, 137, 139, 141, 142, 143, 145, 146, 148, 149, 151, 156, 158, 163, 167, 169, 172, 173, 174, 178, 181, 182, 183, 185, 188, 193

Offset: 1

Gus Wiseman, Jul 25 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.

These are numbers divisible by a prime number not of the form prime(q^k) where q is a prime number and k >= 1.

			The terms together with their prime indices begin:
   13: {6}
   26: {1,6}
   29: {10}
   37: {12}
   39: {2,6}
   43: {14}
   47: {15}
   52: {1,1,6}
   58: {1,10}
   61: {18}
   65: {3,6}
   71: {20}
   73: {21}
   74: {1,12}
   78: {1,2,6}
   79: {22}
   86: {1,14}
   87: {2,10}

Heinz numbers of the partitions counted by A023893.
Allowing prime index 1 gives A356066.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A023894 counts partitions into prime-powers, strict A054685.
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.
A355743 = numbers whose prime indices are prime-powers, squarefree A356065.
Cf. A076610, A085970, A106244, A302492, A302493, A302601, A330946, A354911.

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

A111901 Number of partitions of n into parts that are primes or squares of primes.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 30, 37, 44, 53, 63, 75, 89, 105, 123, 145, 169, 197, 229, 266, 307, 355, 408, 469, 538, 615, 703, 801, 912, 1035, 1175, 1330, 1504, 1698, 1914, 2155, 2423, 2721, 3051, 3418, 3824, 4273, 4770, 5319, 5925

Offset: 0

Reinhard Zumkeller, Aug 20 2005

nonn

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 8*x^10 + ...
a(10) = #{ 7+3, 5+5, 5+3+2, 2^2+2^2+2, 2^2+3+3, 2^2+2+2+2, 3+3+2+2, 2+2+2+2+2 } = 8.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000607, A090677, A023893, A111902.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(tau(d) in [2, 3], d, 0), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 30 2017

a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j]*DivisorSum[j, If[2 <= DivisorSigma[0, #] <= 3, #, 0]&], {j, 1, n}]/n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 06 2017, after Alois P. Heinz *)

{a(n) = if(n < 0, 0, polcoeff( 1 / prod(k=1, primepi(n), (1 - x^prime(k)^2 + x*O(x^n)) * (1 - x^prime(k))), n))}; /* Michael Somos, Dec 26 2016 */

G.f.: Product_{k>=1} 1/((1 - x^prime(k))*(1 - x^(prime(k)^2))). - Ilya Gutkovskiy, Dec 26 2016

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}]

A347643 Number of partitions of n into at most 2 prime powers (including 1).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000961, A023893, A071330, A347644, A347645, A347646, A347647.

A347644 Number of partitions of n into at most 3 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 10, 12, 12, 14, 13, 15, 15, 17, 17, 19, 18, 21, 19, 21, 20, 23, 21, 26, 22, 27, 24, 28, 27, 32, 30, 34, 31, 35, 31, 36, 32, 38, 32, 40, 32, 41, 34, 41, 36, 44, 38, 46, 38, 46, 39, 48, 39, 51, 41, 52, 38, 54, 42, 55, 44, 56

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000961, A023893, A341112, A347643, A347645, A347646, A347647.

A347645 Number of partitions of n into at most 4 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 10, 13, 15, 18, 21, 24, 27, 30, 32, 37, 39, 43, 46, 51, 52, 58, 59, 64, 66, 72, 73, 80, 81, 87, 88, 98, 96, 109, 108, 119, 117, 130, 124, 140, 135, 148, 145, 158, 151, 168, 160, 177, 170, 191, 179, 202, 189, 211, 197, 223, 206, 235, 220, 242

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000961, A023893, A341122, A347643, A347644, A347646, A347647.

A347646 Number of partitions of n into at most 5 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 12, 16, 20, 24, 30, 35, 42, 47, 55, 62, 71, 77, 88, 96, 107, 114, 127, 135, 149, 156, 173, 180, 198, 205, 224, 233, 254, 262, 287, 297, 321, 330, 358, 366, 396, 403, 436, 441, 478, 478, 517, 521, 560, 562, 609, 607, 655, 655, 702, 699, 756, 746

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000961, A023893, A341123, A347643, A347644, A347645, A347647.

A347647 Number of partitions of n into at most 6 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 29, 36, 45, 54, 64, 75, 89, 102, 118, 133, 152, 170, 191, 210, 236, 257, 284, 309, 340, 366, 401, 428, 469, 499, 543, 575, 628, 661, 717, 753, 816, 853, 922, 961, 1035, 1076, 1155, 1195, 1284, 1324, 1417, 1460, 1564, 1604, 1717, 1755

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000961, A023893, A341124, A347643, A347644, A347645, A347646.

cp's OEIS Frontend

A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

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A111901 Number of partitions of n into parts that are primes or squares of primes.

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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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A347643 Number of partitions of n into at most 2 prime powers (including 1).

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A347644 Number of partitions of n into at most 3 prime powers (including 1).

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A347645 Number of partitions of n into at most 4 prime powers (including 1).

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A347646 Number of partitions of n into at most 5 prime powers (including 1).

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A347647 Number of partitions of n into at most 6 prime powers (including 1).

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