A325781 -id:A325781 - cp's OEIS frontend

A366737 Number of numbers k <= A056239(n) that can be written as a linear combination of the prime indices of n (allowing coefficients of 0).

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 4, 1, 5, 4, 4, 1, 5, 1, 5, 3, 6, 1, 5, 2, 7, 3, 6, 1, 6, 1, 5, 5, 8, 4, 6, 1, 9, 4, 6, 1, 7, 1, 7, 6, 10, 1, 6, 2, 7, 6, 8, 1, 7, 4, 7, 5, 11, 1, 7, 1, 12, 4, 6, 3, 8, 1, 9, 7, 8, 1, 7, 1, 13, 7, 10, 4, 9, 1, 7, 4, 14, 1, 8, 5

Offset: 1

Gus Wiseman, Oct 19 2023

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 33 are {2,5}, with combinations
  2 = 2
  4 = 2+2
  5 = 5
  6 = 2+2+2
  7 = 5+2
Hence a(33) = 5.

For minimum instead of length we have A055396.
Positions of first appearances are 1, 2, and A100484.
For subsets instead of combinations we have A304793, complement A325799.
A056239 adds up prime indices, row sums of A112798.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A018818, A046663, A055932, A299701, A365658, A365918, A365921.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Range[Total[prix[n]]],combs[#,prix[n]]!={}&]],{n,30}]

a(2n) = A056239(2n) - 1 for n > 0.

A359420 Numbers that are both practical (A005153) and phi-practical (A260653).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 208, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 272, 280, 288

Offset: 1

Amiram Eldar, Dec 31 2022

nonn

First differs from A325795 at n = 45, and from A325781 at n = 36.

Numbers k such that each number in the range 1..sigma(k) is a sum of distinct divisors of k, and each number in the range 1..k is a subsum of the multiset {phi(d) : d | k}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, Uniform Distribution Theory, Vol. 18, No. 2 (2023), pp. 19-30; arXiv preprint, arXiv:2303.16819 [math.NT], 2023. Mentions this sequence.

Intersection of A005153 and A260653.
Cf. A000010 (phi), A000203 (sigma).
Cf. A325781, A325795.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {};
phiPracticalQ[n_] := If[n == 1, True, (lst = Sort@EulerPhi@Divisors[n]; ok = True; Do[If[lst[[m]] > Sum[lst[[l]], {l, 1, m - 1}] + 1, (ok = False; Break[])], {m, 1, Length[lst]}]; ok)]; (* Frank M Jackson's code at A260653 *)
Select[Range[300], pracQ[#] && phiPracticalQ[#] &]

A366129 Number of finite sets of positive integers with greatest non-subset-sum n.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 7, 11, 11, 15, 18, 23, 28, 36, 40, 50, 59, 70, 83, 101, 118, 141, 166, 195, 227, 268, 306, 358, 414, 478, 549, 640, 730, 846, 968, 1113, 1271, 1462, 1657, 1897, 2154, 2451

Offset: 1

Gus Wiseman, Oct 07 2023

A non-subset-sum of a set summing to n is a positive integer up to n that is not the sum of any subset. For example, the non-subset-sums of {1,3,4} are {2,6}.

			The a(1) = 1 through a(8) = 11 sets:
  {2}  {3}    {4}    {5}      {6}      {7}      {8}        {9}
       {1,3}  {1,4}  {2,3}    {2,4}    {2,5}    {2,6}      {2,7}
                     {1,5}    {1,6}    {3,4}    {3,5}      {3,6}
                     {1,2,5}  {1,2,6}  {1,7}    {1,8}      {4,5}
                                       {1,3,4}  {1,3,5}    {2,3,4}
                                       {1,2,7}  {1,2,8}    {1,9}
                                                {1,2,3,8}  {1,3,6}
                                                           {1,4,5}
                                                           {1,2,9}
                                                           {1,2,3,9}
                                                           {1,2,4,9}

For least instead of greatest: A188431, non-strict A126796 (ranks A325781).
The version counting multisets instead of sets is A366127.
These sets counted by sum are A365924, strict A365831.
A046663 counts partitions without a submultiset summing k, strict A365663.
A325799 counts non-subset-sums of prime indices.
A365923 counts partitions by number of non-subset-sums, strict A365545.
Cf. A006827, A276024, A284640, A304792, A365543, A365658, A365661, A365918, A365920, A365921, A365925.

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n], UnsameQ@@#&&Max@@nmz[#]==n&]],{n,15}]

a(31)-a(42) from Erich Friedman, Nov 13 2024

A367106 Triangle read by rows where T(n,k) is the number of complete length-k integer partitions of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Nov 09 2023

An integer partition of n is complete (ranks A325781) if every integer from 0 to n is the sum of some submultiset of the parts.

			Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  0  1  1
  0  0  0  2  1  1
  0  0  0  1  2  1  1
  0  0  0  1  3  2  1  1
  0  0  0  0  3  3  2  1  1
  0  0  0  0  4  5  3  2  1  1
  0  0  0  0  3  5  5  3  2  1  1
  0  0  0  0  4  8  7  5  3  2  1  1
  0  0  0  0  2  9  9  7  5  3  2  1  1
  0  0  0  0  2 11 12 11  7  5  3  2  1  1
  0  0  0  0  1 11 16 13 11  7  5  3  2  1  1
  0  0  0  0  1 14 21 19 15 11  7  5  3  2  1  1
Row n = 11 counts the following partitions (empty columns not shown):
  6311  62111  611111  5111111  41111111  311111111  2111111111  11111111111
  6221  53111  521111  4211111  32111111  221111111
  5321  52211  431111  3311111  22211111
  4421  44111  422111  3221111
        43211  332111  2222111
        42221  322211
        33311  222221
        33221

Column k appears to have A000325(k) nonzero terms.
Column sums are A003513.
Central column T(2n,n) is A007042.
Row sums are A126796, ranks A325781.
The strict case is too sparse, row sums A188431 (complement A365831).
Grouping by maximum instead of length gives A261036.
A000041 counts integer partitions.
A108917 counts knapsack partitions, ranks A299702.
A299701 counts subset-sums of prime indices, firsts A259941.
A365924 counts incomplete partitions, ranks A365830.
Cf. A002033, A018818, A046663, A047967, A055932, A276024, A304792, A365921.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n,{k}],nmz[#]=={}&]],{n,0,15},{k,0,n}]

A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 5, 3, 2, 1, 4, 7, 2, 1, 1, 6, 7, 6, 2, 1, 6, 10, 6, 7, 1, 7, 12, 11, 8, 3, 1, 6, 16, 11, 17, 3, 2, 1, 10, 14, 20, 19, 10, 2, 1, 1, 7, 22, 17, 31, 14, 7, 2, 1, 9, 22, 27, 37, 22, 11, 6, 1, 10, 24, 27, 51, 32, 16, 15

Offset: 0

Gus Wiseman, Nov 19 2023

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			Triangle begins:
  1
  1  1
  1  2
  1  3  1
  1  3  3
  1  5  3  2
  1  4  7  2  1
  1  6  7  6  2
  1  6 10  6  7
  1  7 12 11  8  3
  1  6 16 11 17  3  2
  1 10 14 20 19 10  2  1
  1  7 22 17 31 14  7  2
  1  9 22 27 37 22 11  6
  1 10 24 27 51 32 16 15
  1 11 27 39 57 43 27 22  4
  1  9 33 34 79 57 36 39  7  2
  1 13 31 51 86 77 45 62 14  4  1
Row n = 9 counts the following partitions:
  (9)  (81)         (711)       (621)      (5211)
       (72)         (6111)      (531)      (4311)
       (63)         (522)       (432)      (4221)
       (54)         (51111)     (33111)    (42111)
       (333)        (441)       (222111)   (3321)
       (111111111)  (411111)    (2211111)  (32211)
                    (3222)                 (321111)
                    (3111111)
                    (22221)
                    (21111111)

Row sums are A000041.
Column k = 1 is A088922.
The non-binary version (with zeros) is A365658.
The strict non-binary version (with zeros) is A365832.
The corresponding rank statistic is A366739.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366738 counts semi-sums of partitions, non-binary A304792.
A366741 counts semi-sums of strict partitions, non-binary A365925.
Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.

DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n,10},{k,0,n}],0,2]

cp's OEIS Frontend

A366737 Number of numbers k <= A056239(n) that can be written as a linear combination of the prime indices of n (allowing coefficients of 0).

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A359420 Numbers that are both practical (A005153) and phi-practical (A260653).

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A366129 Number of finite sets of positive integers with greatest non-subset-sum n.

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A367106 Triangle read by rows where T(n,k) is the number of complete length-k integer partitions of n.

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A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

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