A294788 -id:A294788 - cp's OEIS frontend

A291686 Numbers whose prime indices other than 1 are distinct prime numbers.

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 17, 20, 22, 24, 30, 31, 32, 33, 34, 40, 41, 44, 48, 51, 55, 59, 60, 62, 64, 66, 67, 68, 80, 82, 83, 85, 88, 93, 96, 102, 109, 110, 118, 120, 123, 124, 127, 128, 132, 134, 136, 155, 157, 160, 164, 165, 166, 170, 176, 177

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			9 is not in the sequence because the prime indices of 9 = prime(2)*prime(2) are {2,2} which are prime numbers but not distinct.
15 is in the sequence because the prime indices of 15 = prime(2)*prime(3) are {2,3} which are distinct prime numbers.
21 is not in the sequence because the prime indices of 21 = prime(2)*prime(4) are {2,4} which are distinct but not all prime numbers.
24 is in the sequence because the prime indices of 24 = prime(1)*prime(1)*prime(1)*prime(2) are {1,1,1,2} which without the 1s are distinct prime numbers.

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

Cf. A000009, A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A279790, A281113, A294788, A294788, A301750, A302242, A302243.

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

ok(n)={my(t=n>>valuation(n,2)); issquarefree(t) && !#select(p->!isprime(primepi(p)), factor(t)[,1])} \\ Andrew Howroyd, Aug 26 2018

Sum_{n>=1} 1/a(n) = 2 * Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021

A383311 Number of ways to choose a set multipartition (multiset of sets) of a factorization of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 16, 2, 3, 4, 7, 1, 12, 1, 12, 3, 3, 3, 20, 1, 3, 3, 16, 1, 12, 1, 7, 7, 3, 1, 33, 2, 7, 3, 7, 1, 16, 3, 16, 3, 3, 1, 34, 1, 3, 7, 22, 3, 12, 1, 7, 3, 12, 1, 49, 1, 3, 7, 7, 3, 12, 1, 33, 7, 3

Offset: 1

Gus Wiseman, Apr 28 2025

nonn

First differs from A296119 at a(36) = 20, A296119(36) = 21.

			The a(36) = 20 choices are:
  {{2,3,6}}  {{2,3},{2,3}}  {{2},{3},{2,3}}  {{2},{2},{3},{3}}
  {{2,18}}   {{2},{2,9}}    {{2},{2},{9}}
  {{3,12}}   {{2},{3,6}}    {{2},{3},{6}}
  {{4,9}}    {{3},{2,6}}    {{3},{3},{4}}
  {{36}}     {{6},{2,3}}
             {{2},{18}}
             {{3},{3,4}}
             {{3},{12}}
             {{4},{9}}
             {{6},{6}}

The case of a unique choice (positions of 1) is A008578.
For multisets of multisets we have A050336.
For sets of sets we have A050345.
For normal multisets we have A116540, strong A330783.
For integer partitions instead of factorizations we have A089259.
Twice-partitions of this type are counted by A270995.
For sets of multisets we have A383310 (distinct products A296118).
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A281113 counts twice-factorizations, see A294788, A296120, A296121.
A302478 gives MM-numbers of set multipartitions.
A302494 gives MM-numbers of sets of sets.
Cf. A000009, A001970, A005117, A050342, A116539, A279785, A293243, A293511, A296119, A316439, A381992, A382077.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
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[Sum[Length[Select[mps[y], And@@UnsameQ@@@#&]], {y,facs[n]}],{n,100}]

cp's OEIS Frontend

A291686 Numbers whose prime indices other than 1 are distinct prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula