A357980 -id:A357980 - cp's OEIS frontend

A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172

Offset: 1

Gus Wiseman, Aug 27 2025

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.

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

The complement for all partitions appears to be A276078, counted by A052335.
For all partitions we appear to have A276079, counted by A387134.
For divisors instead of strict partitions we have A355740, counted by A370320.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873, see A387111.
For initial intervals instead of strict partitions we have A387113, counted by A387118.
These are the positions of 0 in A387115.
Partitions of this type are counted by A387137, complement A387178.
The complement is A387177.
The version for constant partitions is A387180, counted by A387329.
The complement for constant partitions is A387181, counted by A387330.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A000720, A261049, A270995, A335433, A335448, A355744, A357978, A357980, A383706, A387110, A387120.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[#]],UnsameQ@@#&]=={}&]

A357853 Fully multiplicative with a(prime(k)) = A000009(k+1).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 4, 2, 5, 3, 4, 1, 6, 4, 8, 2, 6, 4, 10, 2, 4, 5, 8, 3, 12, 4, 15, 1, 8, 6, 6, 4, 18, 8, 10, 2, 22, 6, 27, 4, 8, 10, 32, 2, 9, 4, 12, 5, 38, 8, 8, 3, 16, 12, 46, 4, 54, 15, 12, 1, 10, 8, 64, 6, 20, 6, 76, 4, 89, 18, 8, 8, 12, 10

Offset: 1

Gus Wiseman, Oct 28 2022

			We have 525 = prime(2)*prime(3)*prime(3)*prime(4) so a(525) = Q(3)*Q(4)*Q(4)*Q(5) = 2*2*2*3 = 24, where Q = A000009.

Other multiplicative sequences: A003961, A064988, A064989, A357852, A357980.
The non-strict version is A003964.
The unshifted horizontal version is A357978, non-strict A357977.
The unshifted version is A357982.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000009, A000720, A076610, A273873, A296150, A357979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ptf[f_][n_]:=Product[f[i],{i,primeMS[n]}];
Array[ptf[PartitionsQ[#+1]&],100]

A357984 Replace prime(k) with A000720(k) in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 25 2022

			We have 91 = prime(4) * prime(6), so a(91) = pi(4) * pi(6) = 6.

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357983.
The version for p instead of pi is A299200, horz A357977, strict A357982.
The version for nu is A355741.
The version for bigomega is A355742.
The horizontal version is A357980.
A000040 lists the prime numbers.
A000720 is PrimePi.
A056239 adds up prime indices, row-sums of A112798.
Cf. A033879, A033880, A063834, A066207, A076610, A296150, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@PrimePi/@primeMS[n],{n,100}]

A357981 Numbers whose prime indices have only prime numbers as their own prime indices.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 11, 16, 20, 22, 23, 25, 31, 32, 40, 44, 46, 47, 50, 55, 59, 62, 64, 80, 88, 92, 94, 97, 100, 103, 110, 115, 118, 121, 124, 125, 127, 128, 137, 155, 160, 176, 179, 184, 188, 194, 197, 200, 206, 220, 230, 233, 235, 236, 242, 248, 250, 253, 254

Offset: 1

Gus Wiseman, Oct 23 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.

Numbers whose prime indices are prime numbers are listed by A076610.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     5: {3}
     8: {1,1,1}
    10: {1,3}
    11: {5}
    16: {1,1,1,1}
    20: {1,1,3}
    22: {1,5}
    23: {9}
    25: {3,3}
    31: {11}
    32: {1,1,1,1,1}

Contains all elements of A000079.
Contains all primes indexed by elements of A076610.
A000040 lists the prime numbers.
A056239 adds up prime indices, row-sums of A112798.
Cf. A003961, A045966, A064988, A066207, A215366, A357977, A357980, A357983.

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

cp's OEIS Frontend

A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

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A357853 Fully multiplicative with a(prime(k)) = A000009(k+1).

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A357984 Replace prime(k) with A000720(k) in the prime factorization of n.

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A357981 Numbers whose prime indices have only prime numbers as their own prime indices.

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