A318283 -id:A318283 - cp's OEIS frontend

A363526 Number of integer partitions of n with reverse-weighted sum 3*n.

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 3, 2, 4, 4, 4, 5, 5, 4, 7, 7, 5, 8, 7, 6, 11, 9, 8, 11, 10, 10, 13, 12, 11, 15, 15, 12, 17, 16, 14, 20, 18, 16, 22, 20, 19, 24, 22, 20, 27, 26, 23, 29, 27, 25, 33, 30, 28, 35, 33, 31, 38, 36, 33, 41, 40

Offset: 0

Gus Wiseman, Jun 10 2023

nonn

Are the partitions counted all of length 4 or 5?

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.

			The partition (6,4,4,1) has sum 15 and reverse-weighted sum 45 so is counted under a(15).
The a(n) partitions for n = {5, 10, 15, 16, 21, 24}:
  (1,1,1,1,1)  (4,3,2,1)    (6,4,4,1)    (6,5,4,1)  (8,6,6,1)   (9,7,7,1)
               (2,2,2,2,2)  (6,5,2,2)    (6,6,2,2)  (8,7,4,2)   (9,8,5,2)
                            (7,3,3,2)    (7,4,3,2)  (9,5,5,2)   (9,9,3,3)
                            (3,3,3,3,3)             (9,6,3,3)   (10,6,6,2)
                                                    (10,4,4,3)  (10,7,4,3)
                                                                (11,5,5,3)
                                                                (12,4,4,4)

Positions of terms with omega > 4 appear to be A079998.
The version for compositions is A231429.
The non-reverse version is A363527.
These partitions have ranks A363530, reverse A363531.
A000041 counts integer partitions, strict A000009.
A053632 counts compositions by weighted sum, rank statistic A029931/A359042.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A000016, A008284, A067538, A222855, A222970, A359755, A360672, A360675, A362559, A362560, A363525, A363528.

Table[Length[Select[IntegerPartitions[n],Total[Accumulate[#]]==3n&]],{n,0,30}]

A318371 Number of non-isomorphic strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 24 2018

			Non-isomorphic representatives of the a(24) = 6 strict set multipartitions of {1,1,2,3,4}:
  {{1},{1,2,3,4}}
  {{1,2},{1,3,4}}
  {{1},{2},{1,3,4}}
  {{1},{1,2},{3,4}}
  {{2},{1,3},{1,4}}
  {{1},{2},{3},{1,4}}

Cf. A007716, A045778, A049311, A050326, A116539, A116540, A283877, A292432 A292444.

Cf. A318283, A318285, A318287, A318360, A318361, A318362, A318369, A318370.

a(n) = A318370(A181821(n)).

A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 3, 8, 7, 7, 8, 11, 12, 15, 5, 15, 17, 22, 14, 27, 19, 30, 13, 27, 30, 33, 26, 42, 37, 56, 7, 44, 45, 51, 34, 77, 67, 72, 25

Offset: 1

Gus Wiseman, Aug 28 2018

A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A (headless) combinatory separation of a multiset m is a multiset of normal multisets {t_1,...,t_k} such that there exist multisets {s_1,...,s_k} with multiset union m and such that s_i has type t_i for each i = 1...k.

The prime indices of n are the n-th row of A296150.

			The a(18) = 17 combinatory separations of {1,1,2,2,3}:
  {11223}
  {1,1122} {1,1123} {1,1223} {11,112} {12,112} {12,122} {12,123}
  {1,1,112} {1,1,122} {1,1,123} {1,11,11} {1,11,12} {1,12,12}
  {1,1,1,11} {1,1,1,12}
  {1,1,1,1,1}

Cf. A007716, A056239, A255906, A269134, A296150, A305936.

Cf. A317791, A318283, A318284, A318285, A318559, A318562, A318566, A318567.

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}]]]]];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Union[Sort/@Map[normize,mps[nrmptn[n]],{2}]]],{n,20}]

A363527 Number of integer partitions of n with weighted sum 3*n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 3, 4, 4, 6, 8, 7, 10, 13, 13, 21, 25, 24, 37, 39, 40, 58, 63, 72, 94, 106, 118, 144, 165, 181, 224, 256, 277, 341, 387, 417, 504, 560, 615, 743, 818, 899, 1066, 1171, 1285, 1502, 1655, 1819, 2108, 2315, 2547, 2915

Offset: 0

Gus Wiseman, Jun 11 2023

nonn

Are the partitions counted all of length > 4?

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.

			The partition (2,2,1,1,1,1) has sum 8 and weighted sum 24 so is counted under a(8).
The a(13) = 1 through a(18) = 8 partitions:
  (332221)  (333221)    (33333)     (442222)    (443222)    (443331)
            (4322111)   (522222)    (5322211)   (4433111)   (444222)
            (71111111)  (4332111)   (55111111)  (5332211)   (533322)
                        (63111111)  (63211111)  (55211111)  (4443111)
                                                (63311111)  (7222221)
                                                (72221111)  (55311111)
                                                            (64221111)
                                                            (A11111111)

The version for compositions is A231429.
The reverse version is A363526.
These partitions have ranks A363531.
A000041 counts integer partitions, strict A000009.
A053632 counts compositions by weighted sum, rank statistic A029931/A359042.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A000016, A008284, A067538, A222855, A222970, A359755, A360672, A360675, A362559, A362560, A363525, A363528, A363532.

Table[Length[Select[IntegerPartitions[n],Total[Accumulate[Reverse[#]]]==3n&]],{n,0,30}]

A363530 Heinz numbers of integer partitions such that 3*(sum) = (weighted sum).

Original entry on oeis.org

1, 32, 40, 60, 100, 126, 210, 243, 294, 351, 550, 585, 770, 819, 1210, 1274, 1275, 1287, 1521, 1785, 2002, 2366, 2793, 2805, 2875, 3125, 3315, 4025, 4114, 4335, 4389, 4862, 5187, 6325, 6358, 6422, 6783, 7105, 7475, 7581, 8349, 8398, 9386, 9775, 9867, 10925

Offset: 1

Gus Wiseman, Jun 12 2023

nonn

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.

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18.

			The terms together with their prime indices begin:
      1: {}
     32: {1,1,1,1,1}
     40: {1,1,1,3}
     60: {1,1,2,3}
    100: {1,1,3,3}
    126: {1,2,2,4}
    210: {1,2,3,4}
    243: {2,2,2,2,2}
    294: {1,2,4,4}
    351: {2,2,2,6}
    550: {1,3,3,5}
    585: {2,2,3,6}
    770: {1,3,4,5}
    819: {2,2,4,6}

These partitions are counted by A363527.
The reverse version is A363531, counted by A363526.
A053632 counts compositions by weighted sum.
A055396 gives minimum prime index, maximum A061395.
A112798 lists prime indices, length A001222, sum A056239.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A000041, A000720, A001221, A046660, A106529, A118914, A124010, A181819, A215366, A359362, A359755.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],3*Total[prix[#]]==Total[Accumulate[Reverse[prix[#]]]]&]

A056239(a(n)) = A304818(a(n))/3.

A358102 Numbers of the form prime(w)prime(x)prime(y) with w >= x >= y such that 2w = 3x + 4y.

Original entry on oeis.org

66, 153, 266, 609, 806, 1295, 1599, 1634, 2107, 3021, 3055, 3422, 5254, 5369, 5795, 5829, 7138, 8769, 9443, 9581, 10585, 10706, 12337, 12513, 13298, 16465, 16511, 16849, 17013, 18602, 21983, 22145, 23241, 23542, 26159, 29014, 29607, 29945, 30943, 32623, 32809

Offset: 1

Gus Wiseman, Nov 02 2022

nonn

Also Heinz numbers of integer partitions (w,x,y) summing to n such that 2w = 3x + 4y, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The terms together with their prime indices begin:
     66: {1,2,5}
    153: {2,2,7}
    266: {1,4,8}
    609: {2,4,10}
    806: {1,6,11}
   1295: {3,4,12}
   1599: {2,6,13}
   1634: {1,8,14}
   2107: {4,4,14}
   3021: {2,8,16}
   3055: {3,6,15}
   3422: {1,10,17}
   5254: {1,12,20}
   5369: {4,6,17}
   5795: {3,8,18}
   5829: {2,10,19}
   7138: {1,14,23}
   8769: {2,12,22}

The ordered version is A357489, apparently counted by A008676.
These partitions are counted by A357849.
A000040 lists the primes.
A000041 counts partitions, strict A000009.
A003963 multiplies prime indices.
A056239 adds up prime indices.
Cf. A000720, A001221, A001222, A215366, A296150, A318283.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],PrimeOmega[#]==3&&2*primeMS[#][[-1]]==3*primeMS[#][[-2]]+4*primeMS[#][[-3]]&]

A359397 Squarefree numbers with weakly decreasing first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 19, 21, 23, 29, 30, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 127, 131, 133, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 187, 191, 193, 197

Offset: 1

Gus Wiseman, Dec 31 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.

			715 has prime indices {3,5,6}, with first differences (2,1), which are weakly decreasing, so 715 is in the sequence.

This is the squarefree case of A325362.
These are the sorted Heinz numbers of rows of A359361.
A005117 lists squarefree numbers.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A355536 lists first differences of prime indices, 0-prepended A287352.
A358136 lists partial sums of prime indices, row sums A318283.
Cf. A000009, A000720, A001221, A253566, A261079, A304818, A358137, A358169.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&GreaterEqual@@Differences[Prepend[primeMS[#],0]]&]

Intersection of A325362 and A005117.

A363525 Number of integer partitions of n with weighted sum divisible by reverse-weighted sum.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 5, 5, 3, 10, 4, 7, 13, 10, 8, 29, 10, 18, 39, 20, 20, 70, 29, 40, 105, 65, 55, 166, 73, 132, 242, 141, 129, 476, 183, 248, 580, 487, 312, 984, 422, 868, 1345, 825, 724, 2709, 949, 1505, 2756, 2902, 1611, 4664, 2289, 4942, 5828, 4278

Offset: 1

Gus Wiseman, Jun 10 2023

nonn

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.

			The partition (6,5,4,3,2,1,1,1,1) has weighted sum 80, reverse 160, so is counted under a(24).
The a(n) partitions for n = 1, 2, 4, 6, 9, 12, 14 (A..E = 10-14):
  1  2   4     6       9          C             E
     11  22    33      333        66            77
         1111  222     711        444           65111
               111111  6111       921           73211
                       111111111  3333          2222222
                                  7311          71111111
                                  63111         11111111111111
                                  222222
                                  621111
                                  111111111111

The case of equality (and reciprocal version) is A000005.
The strict case is A363528.
A000041 counts integer partitions, strict A000009.
A053632 counts compositions by weighted sum, rank statistic A029931/A359042.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
A363526 = partitions with weighted sum 3n, ranks A363530, reverse A363531.
Cf. A000016, A008284, A067538, A222855, A222970, A358137, A359755, A362558, A362559, A362560, A363527.

Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[#]], Total[Accumulate[Reverse[#]]]]&]],{n,30}]

A363528 Number of strict integer partitions of n with weighted sum divisible by reverse-weighted sum.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 6, 2, 3, 9, 3, 4, 11, 4, 5, 16, 6, 8, 24, 8, 10, 31, 11, 14, 41, 18, 18, 59, 21, 27, 74, 30, 32, 100, 35, 43, 128, 54, 53, 173, 58, 78, 215, 81, 88, 294, 97, 123, 362, 150, 146, 469, 162, 221, 577

Offset: 1

Gus Wiseman, Jun 10 2023

nonn

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.

			The a(n) partitions for n = 1, 12, 15, 21, 24, 26:
  (1)  (12)     (15)       (21)          (24)          (26)
       (9,2,1)  (11,3,1)   (15,5,1)      (17,6,1)      (11,8,4,2,1)
                (9,3,2,1)  (16,3,2)      (18,4,2)      (12,6,5,2,1)
                           (11,7,2,1)    (12,9,2,1)    (13,5,4,3,1)
                           (12,5,3,1)    (13,7,3,1)
                           (10,5,3,2,1)  (14,5,4,1)
                                         (15,4,3,2)
                                         (10,8,3,2,1)
                                         (11,6,4,2,1)

The non-strict version is A363525.
A000041 counts integer partitions, strict A000009.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
A363526 counts partitions with weighted sum 3n, reverse A363531.
Cf. A008284, A053632, A067538, A222855, A222970, A358137, A359754, A359755, A362558, A362559, A362560.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Divisible[Total[Accumulate[#]],Total[Accumulate[Reverse[#]]]]&]],{n,30}]

A359757 Greatest positive integer whose weakly increasing prime indices have zero-based weighted sum (A359674) equal to n.

Original entry on oeis.org

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 12167, 11449, 15341, 24389, 16399, 26071, 29791, 31117, 35557, 50653, 39401, 56129, 68921, 58867, 72283, 83521, 79007, 86903, 103823

Offset: 1

Gus Wiseman, Jan 16 2023

nonn

Appears to first differ from A001248 at a(27) = 12167, A001248(27) = 10609.
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 zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

			The terms together with their prime indices begin:
    4: {1,1}
    9: {2,2}
   25: {3,3}
   49: {4,4}
  121: {5,5}
  169: {6,6}
  289: {7,7}
  361: {8,8}
  529: {9,9}
  841: {10,10}

Andrew Howroyd, Table of n, a(n) for n = 1..500

The one-based version is A359497, minimum A359682 (sorted A359755).
Last position of n in A359674, reverse A359677.
The minimum instead of maximum is A359676, sorted A359675, reverse A359681.
A053632 counts compositions by zero-based weighted sum.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124757 = zero-based weighted sum of standard compositions, reverse A231204.
A304818 gives weighted sums of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 = partial sums of prime indices, ranked by A358137, reverse A359361.
Cf. A001248, A029931, A055932, A089633, A243055, A358194, A359679, A359680, A359683, A359754.

nn=10;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
seq=Table[wts[prix[n]],{n,2^nn}];
Table[Position[seq,k][[-1,1]],{k,nn}]

a(n)={ my(recurse(r, k, m) = if(k==1, if(m>=r, prime(r)^2),
    my(z=0); for(j=1, min(m, (r-k*(k-1)/2)\k), z=max(z, self()(r-k*j, k-1, j)*prime(j))); z));
  vecmax(vector((sqrtint(8*n+1)-1)\2, k, recurse(n,k,n)));
} \\ Andrew Howroyd, Jan 21 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 21 2023

cp's OEIS Frontend

A363526 Number of integer partitions of n with reverse-weighted sum 3*n.

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A318371 Number of non-isomorphic strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

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A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

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A363527 Number of integer partitions of n with weighted sum 3*n.

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A363530 Heinz numbers of integer partitions such that 3*(sum) = (weighted sum).

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A358102 Numbers of the form prime(w)*prime(x)*prime(y) with w >= x >= y such that 2w = 3x + 4y.

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A359397 Squarefree numbers with weakly decreasing first differences of 0-prepended prime indices.

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A363525 Number of integer partitions of n with weighted sum divisible by reverse-weighted sum.

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A363528 Number of strict integer partitions of n with weighted sum divisible by reverse-weighted sum.

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A358102 Numbers of the form prime(w)prime(x)prime(y) with w >= x >= y such that 2w = 3x + 4y.