A299201 -id:A299201 - cp's OEIS frontend

A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.

1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 17, 12, 31, 22, 15, 16, 47, 18, 79, 20, 33, 34, 113, 24, 25, 62, 27, 44, 181, 30, 263, 32, 51, 94, 55, 36, 389, 158, 93, 40, 547, 66, 761, 68, 45, 226, 1049, 48, 121, 50, 141, 124, 1453, 54, 85, 88, 237, 362, 1951, 60, 2659, 526

Offset: 1

Gus Wiseman, Oct 23 2022

In the definition, taking A000041(k) instead of prime(A000041(k)) gives A299200.

			We have 35 = prime(3) * prime(4), so a(35) = prime(A000041(3)) * prime(A000041(4)) = prime(3) * prime(5) = 55.

Applying the same transformation again gives A357979.
The strict version is A357978.
Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PartitionsP],100]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(numbpart(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

A357852 Replace prime(k) with prime(k+2) in the prime factorization of n.

Original entry on oeis.org

1, 5, 7, 25, 11, 35, 13, 125, 49, 55, 17, 175, 19, 65, 77, 625, 23, 245, 29, 275, 91, 85, 31, 875, 121, 95, 343, 325, 37, 385, 41, 3125, 119, 115, 143, 1225, 43, 145, 133, 1375, 47, 455, 53, 425, 539, 155, 59, 4375, 169, 605, 161, 475, 61, 1715, 187, 1625, 203

Offset: 1

Gus Wiseman, Oct 28 2022

This is the same as A045966 except the first term is 1 instead of 3.

			The terms together with their prime indices begin:
    1: {}
    5: {3}
    7: {4}
   25: {3,3}
   11: {5}
   35: {3,4}
   13: {6}
  125: {3,3,3}
   49: {4,4}
   55: {3,5}
   17: {7}
  175: {3,3,4}
   19: {8}
   65: {3,6}
   77: {4,5}
  625: {3,3,3,3}

Applying the transformation only once gives A003961.
A permutation of A007310.
Other multiplicative sequences: A064988, A064989, A357977, A357980, A357983.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A003964, A066207, A076610, A215366, A296150, A299201, A357979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Product[Prime[i+2],{i,primeMS[n]}],{n,30}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = nextprime(nextprime(f[k,1]+1)+1)); factorback(f); \\ Michel Marcus, Oct 28 2022

from math import prod
from sympy import nextprime, factorint
def A357852(n): return prod(nextprime(p,ith=2)**e for p, e in factorint(n).items()) # Chai Wah Wu, Oct 29 2022

a(n) = A003961(A003961(n)).

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0

Offset: 1

Gus Wiseman, Aug 18 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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
Allowing repeated partitions gives A299200, A357977, A357982, A357978.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
For prime factors instead of partitions see A355741, A355742, A387136.
The disjoint case is A383706.
For initial intervals instead of partitions we have A387111.
The case of strict partitions is A387115.
The case of constant partitions is A387120.
Taking each prime factor (instead of index) gives A387133.
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, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

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

A301364 Regular triangle where T(n,k) is the number of enriched p-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 4, 5, 1, 2, 6, 11, 12, 1, 3, 10, 26, 38, 34, 1, 3, 13, 39, 87, 117, 92, 1, 4, 19, 69, 181, 339, 406, 277, 1, 4, 23, 95, 303, 707, 1198, 1311, 806, 1, 5, 30, 143, 514, 1430, 2970, 4525, 4522, 2500, 1, 5, 35, 184, 762, 2446, 6124, 11627

Offset: 1

Gus Wiseman, Mar 19 2018

An enriched p-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

			Triangle begins:
  1
  1   1
  1   1   2
  1   2   4   5
  1   2   6  11  12
  1   3  10  26  38  34
  1   3  13  39  87 117  92
  1   4  19  69 181 339 406 277
  ...
The T(5,4) = 11 enriched p-trees: (((21)1)1), ((2(11))1), (((11)2)1), ((211)1), ((21)(11)), (((11)1)2), ((111)2), ((21)11), (2(11)1), ((11)21), (2111).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Last entries of each row give A196545. Row sums are A289501.

Cf. A008284, A055277, A063834, A220418, A273866, A273873, A281145, A290261, A299201, A299202, A299203, A300354, A300442, A300443, A301365-A301368.

eptrees[n_]:=Prepend[Join@@Table[Tuples[eptrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[Select[eptrees[n],Count[#,_Integer,{-1}]===k&]],{n,8},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); apply(p->Vecrev(p/y), v)}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018

A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 8, 4, 6, 5, 8, 7, 6, 6, 16, 11, 8, 13, 12, 6, 10, 19, 16, 9, 14, 8, 12, 29, 12, 37, 32, 10, 22, 9, 16, 47, 26, 14, 24, 61, 12, 79, 20, 12, 38, 103, 32, 9, 18, 22, 28, 131, 16, 15, 24, 26, 58, 163, 24, 199, 74, 12, 64, 21, 20, 251, 44, 38

Offset: 1

Gus Wiseman, Oct 24 2022

In the definition, taking A000009(k) instead of prime(A000009(k)) gives A357982.

			We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(1) * prime(1)^2 * prime(2) = 24.

The non-strict version is A357977.
Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357979, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PartitionsQ],100]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(f9(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

A300352 Number of strict trees of weight n with distinct leaves.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 8, 11, 17, 40, 48, 76, 109, 159, 400, 470, 745, 1057, 1576, 2103, 5267, 6022, 9746, 13390, 20099, 26542, 39396, 82074, 101387, 152291, 215676, 308937, 423587, 596511, 799022, 1623311, 1960223, 2947722, 4048704, 5845982, 7794809, 11028888

Offset: 1

Gus Wiseman, Mar 03 2018

nonn

A strict tree of weight n > 0 is either a single node of weight n, or a sequence of two or more strict trees with strictly decreasing weights summing to n.

			The a(8) = 11 strict trees with distinct leaves: 8, (71), ((52)1), ((43)1), (62), ((51)2), (53), ((41)3), (5(21)), (521), (431).

Cf. A000009, A056239, A063834, A196545, A246867, A273873, A281145, A289501, A294018, A294079, A296150, A299201, A299202, A299203, A300353, A300354, A300355.

sps[{}]:={{}};sps[set:{i_,_}]:=
Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
str[q_]:=str[q]=If[Length[q]===1,1,Total[Times@@@Map[str,Select[sps[q],And[Length[#]>1,UnsameQ@@Total/@#]&],{2}]]];
Table[Total[str/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,20}]

a(n) = Sum_{i=1..A000009(n)} A294018(A246867(n,i)).

A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 5, 4, 11, 10, 23, 8, 25, 22, 31, 20, 47, 46, 55, 16, 59, 50, 103, 44, 115, 62, 97, 40, 121, 94, 125, 92, 137, 110, 127, 32, 155, 118, 253, 100, 197, 206, 235, 88, 179, 230, 233, 124, 275, 194, 257, 80, 529, 242, 295, 188, 419, 250, 341, 184, 515, 274

Offset: 1

Gus Wiseman, Oct 24 2022

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 define the MTF-transform as shifting a number's prime indices along a function; see the Mathematica program.

			First, we have
- 4 = prime(1) * prime(1),
- A000040(1) = 2,
- A064988(4) = prime(2) * prime(2) = 9.
Similarly, A064988(3) = 5. Next,
- 35 = prime(3) * prime(4),
- A064988(3) = 5,
- A064988(4) = 9,
- a(35) = prime(5) * prime(9) = 253.

Other multiplicative sequences: A003961, A357852, A064989, A357977, A357980.
Applying the transformation only once gives A064988.
The union is A076610 (numbers whose prime indices are themselves prime).
For partition numbers instead of primes we have A357979.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A003964, A063834, A215366, A296150, A299201, A299202, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[mtf[Prime]],100]

A387120 Number of ways to choose a different constant integer partition of each prime index of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 26 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.

			The prime indices of 90 are {1,2,2,3}, with choices:
  ((1),(2),(1,1),(3))
  ((1),(1,1),(2),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(1,1,1))
so a(90) = 4.

For multiset systems see A355529, set systems A367901.
For not necessarily different choices we have A355731, see A355740.
For divisors instead of constant partitions we have A355739 (also the disjoint case).
For prime factors instead of constant partitions we have A387136.
For all instead of just constant partitions we have A387110, disjoint case A383706.
For initial intervals instead of partitions we have A387111.
For strict instead of constant partitions we have A387115.
Twice partitions of this type are counted by A387179, constant-block case of A296122.
Positions of zero are A387180 (non-choosable), complement A387181 (choosable).
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. A000005, A063834, A261049, A299200, A299201, A335433, A335448, A355741, A357977, A387135.

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

A321470 Number of integer partitions of the n-th triangular number 1 + 2 + ... + n that can be obtained by choosing a partition of each integer from 1 to n and combining.

Original entry on oeis.org

1, 1, 2, 5, 16, 54, 212, 834, 3558, 15394, 69512, 313107, 1474095, 6877031, 32877196

Offset: 0

Gus Wiseman, Nov 11 2018

a(n) is the number of integer partitions finer than (n, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + n ordered by refinement.

a(n+1)/a(n) appears to converge as n -> oo. - Chai Wah Wu, Nov 14 2018

			The a(1) = 1 through a(4) = 16 partitions:
  (1)  (21)   (321)     (4321)
       (111)  (2211)    (32221)
              (3111)    (33211)
              (21111)   (42211)
              (111111)  (43111)
                        (222211)
                        (322111)
                        (331111)
                        (421111)
                        (2221111)
                        (3211111)
                        (4111111)
                        (22111111)
                        (31111111)
                        (211111111)
                        (1111111111)
The partition (222211) is the combination of (22)(21)(2)(1), so is counted under a(4). The partition (322111) is the combination of (22)(3)(11)(1), (31)(21)(2)(1), or (211)(3)(2)(1), so is also counted under a(4).

Cf. A000217, A001970, A002846, A063834, A066723, A173519, A213427, A242422, A261049, A265947, A271619, A299201, A300383, A317141.

Cf. A321467, A321468, A321471, A321472, A321514.

Table[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@Range[1,n]]]],{n,6}]

from collections import Counter
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A321470_gen(): # generator of terms
    aset = {(1,)}
    yield 1
    for n in count(2):
        yield len(aset)
        aset = {tuple(sorted(p+q)) for p in aset for q in (tuple(sorted(Counter(q).elements())) for q in partitions(n))}
A321470_list = list(islice(A321470_gen(),10)) # Chai Wah Wu, Sep 20 2023

a(n) <= A173519(n). - David A. Corneth, Sep 20 2023

a(9)-a(11) from Alois P. Heinz, Nov 12 2018
a(12)-a(13) from Chai Wah Wu, Nov 13 2018
a(14) from Chai Wah Wu, Sep 20 2023

A294018 Number of strict trees whose leaves are the parts of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 3, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 0, 1, 4, 1, 1, 1, 3, 1, 6, 1, 1, 1, 1, 1, 4, 1, 1, 0, 1, 1, 8, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 4, 1, 1, 6, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 13

Offset: 1

Gus Wiseman, Feb 06 2018

nonn

By convention a(1) = 0.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(84) = 8 strict trees: (((42)1)1), (((41)2)1), ((4(21))1), ((421)1), (((41)1)2), ((41)(21)), ((41)21), (4(21)1).

Cf. A000009, A000041, A000720, A001222, A056239, A063834, A196545, A215366, A273873, A281145, A289501, A296150, A299201, A299202, A299203.

nn=120;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
qci[y_]:=qci[y]=If[Length[y]===1,1,Sum[Times@@qci/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y,UnsameQ@@Total/@#]&]}]];
qci/@ptns

A273873(n) = Sum_{i=1..A000041(n)} a(A215366(n,i)).

cp's OEIS Frontend

A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.

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A357852 Replace prime(k) with prime(k+2) in the prime factorization of n.

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A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

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A301364 Regular triangle where T(n,k) is the number of enriched p-trees of weight n with k leaves.

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A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

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A300352 Number of strict trees of weight n with distinct leaves.

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A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

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A387120 Number of ways to choose a different constant integer partition of each prime index of n.

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