A357982 -id:A357982 - cp's OEIS frontend

A384396 Position of first appearance of n in A384389 (proper choices of disjoint strict partitions of each prime index).

1, 5, 11, 13, 17, 19, 62, 23, 111, 29, 123, 31, 129, 217, 37, 141, 106, 41, 159, 391, 118, 43

Offset: 0

Gus Wiseman, Jun 03 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.

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

Positions of first appearances in A384389.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294
A351293 counts non-Look-and-Say partitions, ranks A351295.
Cf. A382912, A383710, A382913, A383708.
Cf. A179009, A279790, A357982, A381454, A382525, A383706, A384321, A384322, A384347, A384349, A384390, A384393.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
lv=Table[Length[pofprop[prix[n]]],{n,100}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[lv,x][[1,1]],{x,0,mnrm[lv+1]-1}]

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}]

A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 4, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 4, 1, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 1, 4, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 4, 2, 1, 2, 1, 3, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 23 2025

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 prime indices of 180 are {1,1,2,2,3}, conjugate {5,3,1}, and we have choices:
  {{5},{3},{1}}
  {{5},{2,1},{1}}
  {{4,1},{3},{1}}
  {{4,1},{2,1},{1}}
  {{3,2},{3},{1}}
  {{3,2},{2,1},{1}}
so a(180) = 6.

Positions of 1 are A037143, complement A033942.
For multiplicities instead of indices we have A050361.
Adding up over all integer partitions gives A270995, disjoint A279790, strict A279375.
The conjugate version is A357982, disjoint A383706.
The disjoint case is A384005.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non Look-and-Say or non section-sum partitions, ranks A351295 or A381433.
Cf. A122111, A130091, A179009, A382913, A383707, A383710, A384010, A384011.

fop[y_]:=Join@@@Tuples[strptns/@y];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[fop[conj[prix[n]]]],{n,100}]

A384723 Heinz numbers of conjugates of maximally refined strict integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 24, 30, 60, 90, 120, 150, 180, 210, 240, 420, 540, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 2520, 3360, 4620, 6300, 6930, 7560, 9240

Offset: 1

Gus Wiseman, Jun 09 2025

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Given a partition, the following are equivalent:
1) The distinct parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    30: {1,2,3}
    60: {1,1,2,3}
    90: {1,2,2,3}
   120: {1,1,1,2,3}
   150: {1,2,3,3}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   240: {1,1,1,1,2,3}
   420: {1,1,2,3,4}
   540: {1,1,2,2,2,3}
   630: {1,2,2,3,4}
   840: {1,1,1,2,3,4}

Partitions of this type are counted by A179009.
The conjugate version is A383707, proper A384390.
Appears to be the positions of 1 in A384005 (conjugate A383706).
For at least one instead of exactly one choice we appear to have A384010.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
A357982 counts strict partitions of prime indices, non-strict A299200.
Cf. A003963, A048767, A130091, A382525, A384317, A384318, A384320, A384347, A384349, A384394.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[100],With[{y=conj[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]=={}]&]

A387116 Number of ways to choose a constant sequence of integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 5, 1, 2, 0, 7, 0, 11, 0, 0, 1, 15, 0, 22, 0, 0, 0, 30, 0, 3, 0, 2, 0, 42, 0, 56, 1, 0, 0, 0, 0, 77, 0, 0, 0, 101, 0, 135, 0, 0, 0, 176, 0, 5, 0, 0, 0, 231, 0, 0, 0, 0, 0, 297, 0, 385, 0, 0, 1, 0, 0, 490, 0, 0, 0, 627, 0, 792, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Aug 21 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.

If n is a prime power prime(x)^y, then a(n) is the number of integer partitions of x; otherwise, a(n) = 0.

			The a(49) = 5 choices:
  ((4),(4))
  ((3,1),(3,1))
  ((2,2),(2,2))
  ((2,1,1),(2,1,1))
  ((1,1,1,1),(1,1,1,1))

Positions of zeros are A024619, complement A000961.
Twice-partitions of this type are counted by A047968, see also A296122.
For initial intervals instead of partitions we have A055396, see also A387111.
This is the constant case of A299200, see also A357977, A357982.
For disjoint instead of constant we have A383706.
For distinct instead of constant we have A387110.
For divisors instead of partitions we have A387114, see also A355731, A355739.
For strict partitions instead of partitions we have A387117.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
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. A008284, A063834, A276078, A335433, A335448, A355529, A355741, A387115, A387120, A387133, A387136.

Table[Switch[n,1,1,?PrimePowerQ,PartitionsP[PrimePi[FactorInteger[n][[1,1]]]],,0],{n,100}]

a(n) = A000041(A297109(n)).

cp's OEIS Frontend

A384396 Position of first appearance of n in A384389 (proper choices of disjoint strict partitions of each prime index).

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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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A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.

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A384723 Heinz numbers of conjugates of maximally refined strict integer partitions.

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A387116 Number of ways to choose a constant sequence of integer partitions, one of each prime index of n.

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