A321472 -id:A321472 - cp's OEIS frontend

A242422 Numbers in whose prime factorization the indices of primes sum to a triangular number.

1, 2, 5, 6, 8, 13, 21, 22, 25, 27, 28, 29, 30, 36, 40, 46, 47, 48, 57, 64, 73, 76, 85, 86, 91, 102, 107, 117, 121, 123, 130, 136, 142, 147, 151, 154, 156, 164, 165, 175, 185, 189, 196, 197, 198, 201, 206, 208, 210, 217, 220, 222, 225, 243, 250, 252, 257, 264, 268, 270, 279, 280, 296, 298, 299, 300

Offset: 1

Antti Karttunen, May 16 2014

nonn

Numbers k such that A010054(A056239(k)) is one, or equally, that A002262(A056239(k)) is zero.
In "Bulgarian solitaire" a deck of cards or another finite set of objects is divided into one or more piles, and the "Bulgarian operation" is performed by taking one card from each pile, and making a new pile of them. The question originally posed was: on what condition the resulting partitions will eventually reach a fixed point, that is, a collection of piles that will be unchanged by the operation. See Martin Gardner reference and the Wikipedia-page.
This sequence answers the question when we implement the operation on the partition list A112798: These are all such numbers that starting iterating A242424 from them leads eventually to a fixed point, which will be one of the primorial numbers, A002110.
Contains the same terms as rows of A215366 indexed with triangular numbers (A000217: 0, 1, 3, 6, ...), although not in the same order. {1}, {2}, {5, 6, 8}, {13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64}, etc.
Heinz numbers of integer partitions of triangular numbers. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). - Gus Wiseman, Nov 13 2018

			1 is present as it has an empty factorization, for which the sum of prime indices is zero, and zero is also a triangular number.
2 = p_1 is present as 1 is a triangular number.
6 = p_1 * p_2 is present, as 1+2 = 3 is a triangular number.
300 = 2*2*3*5*5 = p_1 * p_1 * p_2 * p_3 * p_3 is present, as 1+1+2+3+3 = 10 is a triangular number.
Any primorial number p_1 * p_2 * p_3 * ... * p_n is present, as 1+2+3+...+n is by definition a triangular number.
The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (), (1), (3), (2,1), (1,1,1), (6), (4,2), (5,1), (3,3), (2,2,2), (4,1,1), (10), (3,2,1), (2,2,1,1), (3,1,1,1), (9,1), (15), (2,1,1,1,1), (8,2), (1,1,1,1,1,1), (21), (8,1,1), (7,3), (14,1), (6,4). - _Gus Wiseman_, Nov 13 2018

Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.

Antti Karttunen, Table of n, a(n) for n = 1..2455
Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).
Wikipedia, Bulgarian solitaire

Complement: A242423.
A002110 (primorial numbers) is a subsequence.
Cf. A112798, A215366, A242424, A000217, A002262, A010054, A056239, A226062, A037481.
Cf. A066723, A321470, A321471, A321472.

triQ[n_]:=Module[{k,i},For[k=n;i=1,k>0,i++,k-=i];k==0];
Select[Range[100],triQ[Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]]&] (* Gus Wiseman, Nov 13 2018 *)

A321468 Number of factorizations of n! into factors > 1 that can be obtained by taking the multiset union of a choice of factorizations of each positive integer from 2 to n into factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 10, 20, 40, 40, 116, 116, 232, 464, 1440, 1440, 4192, 4192, 11640, 23280, 46560, 46560, 157376

Offset: 0

Gus Wiseman, Nov 11 2018

a(n) is the number of factorizations finer than (2*3*...*n) in the poset of factorizations of n! into factors > 1, ordered by refinement.

			The a(2) = 1 through a(8) = 10 factorizations:
2  2*3  2*3*4    2*3*4*5    2*3*4*5*6      2*3*4*5*6*7      2*3*4*5*6*7*8
        2*2*2*3  2*2*2*3*5  2*2*2*3*5*6    2*2*2*3*5*6*7    2*2*2*3*5*6*7*8
                            2*2*3*3*4*5    2*2*3*3*4*5*7    2*2*3*3*4*5*7*8
                            2*2*2*2*3*3*5  2*2*2*2*3*3*5*7  2*2*3*4*4*5*6*7
                                                            2*2*2*2*3*3*5*7*8
                                                            2*2*2*2*3*4*5*6*7
                                                            2*2*2*3*3*4*4*5*7
                                                            2*2*2*2*2*2*3*5*6*7
                                                            2*2*2*2*2*3*3*4*5*7
                                                            2*2*2*2*2*2*2*3*3*5*7
For example, 2*2*2*2*2*2*3*5*6*7 = (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2), so (2*2*2*2*2*2*3*5*6*7) is counted under a(8).

Dominated by A321514.

Cf. A001055, A066723, A076716, A157612, A242422, A265947, A300383, A317144, A317145, A317534, A321467, A321470, A321471, A321472.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Union[Sort/@Join@@@Tuples[facs/@Range[2,n]]]],{n,10}]

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

A321471 Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for some k.

Original entry on oeis.org

2, 6, 8, 30, 36, 40, 48, 64, 210, 252, 270, 280, 300, 324, 336, 360, 400, 432, 448, 480, 576, 640, 768, 1024, 2310, 2772, 2940, 2970, 3080, 3150, 3300, 3528, 3564, 3696, 3780, 3920, 3960, 4050, 4200, 4400, 4500, 4536, 4704, 4752, 4860, 4928, 5040, 5280, 5400

Offset: 1

Gus Wiseman, Nov 13 2018

nonn

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

These partitions are those that are finer than (k, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + k, for some k, ordered by refinement.

			The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (1), (21), (111), (321), (2211), (3111), (21111), (111111), (4321), (42211), (32221), (43111), (33211), (222211), (421111), (322111), (331111), (2221111), (4111111), (3211111), (22111111), (31111111), (211111111), (1111111111).
The partition (322111) has Heinz number 360 and can be partitioned as ((1)(2)(3)(112)), ((1)(2)(12)(13)), or ((1)(11)(3)(22)), so 360 belongs to the sequence.

Subsequence of A242422.
Cf. A001970, A002846, A056239, A066723, A112798, A213427, A242422, A265947, A300383, A317141.
Cf. A321467, A321468, A321470, A321472, A321514.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[2,1000],Select[Map[Total[primeMS[#]]&,facs[#],{2}],Sort[#]==Range[Max@@#]&]!={}&]

A321467 Number of factorizations of n! into factors > 1 that can be obtained by taking the block-products of some set partition of {2,3,...,n}.

Original entry on oeis.org

1, 1, 1, 2, 5, 15, 47, 183, 719, 3329, 14990, 83798, 393864, 2518898

Offset: 0

Gus Wiseman, Nov 11 2018

a(n) is the number of factorizations coarser than (2*3*...*n) in the poset of factorizations of n! into factors > 1, ordered by refinement.

			The a(1) = 1 through a(5) = 15 factorizations:
  ()  (2)  (6)    (24)     (120)
           (2*3)  (3*8)    (2*60)
                  (4*6)    (3*40)
                  (2*12)   (4*30)
                  (2*3*4)  (5*24)
                           (6*20)
                           (8*15)
                           (10*12)
                           (3*5*8)
                           (4*5*6)
                           (2*3*20)
                           (2*4*15)
                           (2*5*12)
                           (3*4*10)
                           (2*3*4*5)
For example, 10*12 = (2*5)*(3*4), so (10*12) is counted under a(5).

Dominated by A000110.

Cf. A001055, A066723, A157612, A242422, A265947, A317141, A317144, A317145, A317534, A321468, A321470, A321471, A321472, A321514.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Union[Sort/@Apply[Times,sps[Range[2,n]],{2}]]],{n,10}]

A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 12, 24, 48, 48, 192, 192, 384, 768, 3840, 3840, 15360, 15360, 61440, 122880, 245760, 245760, 1720320, 3440640, 6881280, 20643840, 82575360, 82575360, 412876800, 412876800, 2890137600, 5780275200, 11560550400, 23121100800, 208089907200

Offset: 1

Gus Wiseman, Nov 11 2018

nonn

			The a(8) = 12 ways to choose a factorization of each integer from 2 to 8:
  (2)*(3)*(4)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*2*2)

Dominates A321468.

Cf. A001055, A050336, A066723, A076716, A157612, A281113, A300383, A317144, A317145, A321467, A321470, A321471, A321472.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Array[Length[facs[#]]&,n,1,Times],{n,30}]

a(n) = Product_{k = 1..n} A001055(k).

A322077 In the ranked poset of integer partitions ordered by refinement, number of integer partitions coarser (greater) than or equal to the integer partition whose multiplicities are the prime indices of n in weakly decreasing order.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 8, 6, 7, 9, 11, 10, 12, 13, 15, 18, 22, 15, 19, 14, 30, 24, 22, 21, 40, 23, 42, 29, 56, 36, 27, 29, 34, 47, 77, 41, 39, 40

Offset: 1

Gus Wiseman, Nov 25 2018

This partition (reversed row n of A305936) is generally not the same as the integer partition with Heinz number n. For example, 12 is the Heinz number of (2,1,1), while the integer partition whose multiplicities are (2,1,1) is (3,2,1,1).

			The list of a(1) = 1 through a(18) = 18 coarser partitions:
  ()  (1)  (2)   (3)   (3)    (4)    (4)     (6)    (6)     (5)     (5)
           (11)  (21)  (21)   (22)   (22)    (33)   (33)    (32)    (32)
                       (111)  (31)   (31)    (42)   (42)    (41)    (41)
                              (211)  (211)   (51)   (51)    (221)   (221)
                                     (1111)  (321)  (222)   (311)   (311)
                                                    (321)   (2111)  (2111)
                                                    (411)           (11111)
                                                    (2211)
.
  (7)     (6)       (6)      (7)      (10)    (7)        (9)
  (43)    (33)      (33)     (43)     (55)    (43)       (54)
  (52)    (42)      (42)     (52)     (64)    (52)       (63)
  (61)    (51)      (51)     (61)     (73)    (61)       (72)
  (322)   (222)     (222)    (322)    (82)    (322)      (81)
  (331)   (321)     (321)    (331)    (91)    (331)      (333)
  (421)   (411)     (411)    (421)    (433)   (421)      (432)
  (511)   (2211)    (2211)   (511)    (442)   (511)      (441)
  (3211)  (3111)    (3111)   (2221)   (532)   (2221)     (522)
          (21111)   (21111)  (3211)   (541)   (3211)     (531)
          (111111)           (4111)   (631)   (4111)     (621)
                             (22111)  (721)   (22111)    (711)
                                      (4321)  (31111)    (3222)
                                              (211111)   (3321)
                                              (1111111)  (4221)
                                                         (4311)
                                                         (5211)
                                                         (32211)

Cf. A063834, A066723, A213427, A265947, A299201, A300383, A317141, A321468, A321472.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[Union[Sort/@Apply[Plus,mps[nrmptn[n]],{2}]]],{n,20}]

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A242422 Numbers in whose prime factorization the indices of primes sum to a triangular number.

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A321468 Number of factorizations of n! into factors > 1 that can be obtained by taking the multiset union of a choice of factorizations of each positive integer from 2 to n into factors > 1.

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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.

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A321471 Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for some k.

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A321467 Number of factorizations of n! into factors > 1 that can be obtained by taking the block-products of some set partition of {2,3,...,n}.

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A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

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A322077 In the ranked poset of integer partitions ordered by refinement, number of integer partitions coarser (greater) than or equal to the integer partition whose multiplicities are the prime indices of n in weakly decreasing order.

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