A327908
Nonprime squarefree numbers whose prime indices all have the same Omega (A001222) and the same sum of prime indices (A056239).
Original entry on oeis.org
1, 667, 2021, 4331, 5767, 7081, 7663, 11021, 11639, 12091, 13837, 15049, 20413, 23213, 26123, 31553, 34933, 35657, 35723, 37909, 37979, 38021, 41449, 42919, 44197, 45113, 51019, 53531, 58339, 63407, 66013, 70531, 72929, 73373, 73903, 75763, 83411, 87361, 90581
Offset: 1
The sequence of terms together with their prime indices begins:
1: {}
667: {9,10}
2021: {14,15}
4331: {18,20}
5767: {21,22}
7081: {21,25}
7663: {22,25}
11021: {27,28}
11639: {27,30}
12091: {28,30}
13837: {26,33}
15049: {26,35}
20413: {33,35}
23213: {34,39}
26123: {36,40}
31553: {34,49}
34933: {42,44}
35657: {42,45}
35723: {34,55}
37909: {39,49}
Equal sum of prime indices:
A327901Equal average of prime indices:
A327902
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primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@PrimeOmega/@primeMS[#]&&SameQ@@Total/@primeMS/@primeMS[#]&]
A371788
Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 8, 4, 1, 0, 2, 19, 24, 6, 1, 0, 2, 47, 95, 49, 9, 1, 0, 6, 105, 363, 297, 93, 12, 1, 0, 12, 248, 1292, 1660, 753, 158, 16, 1, 0, 11, 563, 4649, 8409, 5591, 1653, 250, 20, 1, 0, 2, 1414, 15976, 41264, 38074, 15590, 3249, 380, 25, 1
Offset: 0
The set partition {{1,3},{2},{4}} has two distinct block-sums {2,4} so is counted under T(4,2).
Triangle begins:
1
0 1
0 1 1
0 2 2 1
0 2 8 4 1
0 2 19 24 6 1
0 2 47 95 49 9 1
0 6 105 363 297 93 12 1
0 12 248 1292 1660 753 158 16 1
0 11 563 4649 8409 5591 1653 250 20 1
0 2 1414 15976 41264 38074 15590 3249 380 25 1
Row n = 4 counts the following set partitions:
. {{1,4},{2,3}} {{1},{2,3,4}} {{1},{2},{3,4}} {{1},{2},{3},{4}}
{{1,2,3,4}} {{1,2},{3},{4}} {{1},{2,3},{4}}
{{1,2},{3,4}} {{1},{2,4},{3}}
{{1,3},{2},{4}} {{1,4},{2},{3}}
{{1,3},{2,4}}
{{1,2,3},{4}}
{{1,2,4},{3}}
{{1,3,4},{2}}
A version for integer partitions is
A116608.
For block lengths instead of sums we have
A208437.
A008277 counts set partitions by length.
A275780 counts set partitions with distinct block-sums.
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sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]], Length[Union[Total/@#]]==k&]],{n,0,5},{k,0,n}]
A358835
Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.
Original entry on oeis.org
1, 1, 3, 4, 8, 8, 17, 16, 31, 34, 54, 57, 108, 102, 166, 191, 294, 298, 504, 491, 803, 843, 1251, 1256, 2167, 1974, 3133, 3226, 4972, 4566, 8018, 6843, 11657, 11044, 17217, 15010, 28422, 21638, 38397, 35067, 58508, 44584, 91870, 63262, 125114, 106264, 177483
Offset: 0
The a(1) = 1 through a(6) = 17 multiset partitions:
{1} {2} {3} {4} {5} {6}
{11} {12} {13} {14} {15}
{1}{1} {111} {22} {23} {24}
{1}{1}{1} {112} {113} {33}
{1111} {122} {114}
{2}{2} {1112} {123}
{11}{11} {11111} {222}
{1}{1}{1}{1} {1}{1}{1}{1}{1} {1113}
{1122}
{3}{3}
{11112}
{111111}
{12}{12}
{2}{2}{2}
{111}{111}
{11}{11}{11}
{1}{1}{1}{1}{1}{1}
The version for set partitions is
A327899.
For distinct instead of constant lengths and sums we have
A358832.
The version for twice-partitions is
A358833.
A001970 counts multiset partitions of integer partitions.
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Table[If[n==0,1,Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d],n/d],SameQ@@Length/@#&],{d,Divisors[n]}]]]],{n,0,20}]
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P(n,y) = 1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ Andrew Howroyd, Dec 31 2022
Showing 1-3 of 3 results.