A325758 -id:A325758 - cp's OEIS frontend

A325757 Irregular triangle read by rows giving the frequency span of n.

1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 4, 1, 1, 1, 3, 2, 2, 2, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 2, 2, 6, 1, 1, 1, 2, 4, 1, 1, 2, 2, 3, 1, 1, 1, 1, 4, 7, 1, 1, 1, 1, 2, 2, 2, 2, 8, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 4, 1, 1, 1, 2, 5, 9, 1, 1, 1, 1, 1, 1, 2, 2, 3

Offset: 1

Gus Wiseman, May 19 2019

We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer is the frequency span of its prime indices (row n of A296150).

			Triangle begins:
   1:
   2: 1
   3: 2
   4: 1 1 2
   5: 3
   6: 1 1 1 2 2
   7: 4
   8: 1 1 1 3
   9: 2 2 2
  10: 1 1 1 2 3
  11: 5
  12: 1 1 1 1 1 2 2 2
  13: 6
  14: 1 1 1 2 4
  15: 1 1 2 2 3
  16: 1 1 1 1 4
  17: 7
  18: 1 1 1 1 2 2 2 2
  19: 8
  20: 1 1 1 1 1 2 2 3
  21: 1 1 2 2 4
  22: 1 1 1 2 5
  23: 9
  24: 1 1 1 1 1 1 2 2 3
  25: 2 3 3
  26: 1 1 1 2 6
  27: 2 2 2 3
  28: 1 1 1 1 1 2 2 4

Row lengths are A325249.
Run-lengths are A325758.
Number of distinct terms in row n is A325759(n).
Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A290822, A323014, A324843, A325277, A325755, A325760.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
Table[freqspan[primeMS[n]],{n,15}]

A325759 Number of distinct frequencies in the frequency span of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 3, 2, 1, 2, 1, 3, 3, 3, 1, 3, 2, 3, 2, 3, 1, 3, 1, 2, 3, 3, 4, 2, 1, 3, 3, 3, 1, 4, 1, 3, 3, 3, 1, 3, 2, 3, 3, 3, 1, 3, 4, 4, 3, 3, 1, 3, 1, 3, 3, 2, 4, 4, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 4, 4, 1, 4, 2, 3, 1, 3, 4, 3, 3

Offset: 1

Gus Wiseman, May 19 2019

nonn

We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer n is the frequency span of its prime indices (row n of A296150).

Row lengths of A325758.
Number of distinct entries in row n of A325757.
Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A323014, A325249, A325277, A325760.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
Table[Length[Union[freqspan[primeMS[n]]]],{n,100}]

A325760 Heinz number of the frequency span of n.

Original entry on oeis.org

1, 2, 3, 12, 5, 72, 7, 40, 27, 120, 11, 864, 13, 168, 180, 112, 17, 1296, 19, 1440, 252, 264, 23, 2880, 75, 312, 135, 2016, 29, 1200, 31, 352, 396, 408, 420, 972, 37, 456, 468, 4800, 41, 1680, 43, 3168, 3240, 552, 47, 8064, 147, 3600, 612, 3744, 53, 6480, 660

Offset: 1

Gus Wiseman, May 19 2019

nonn

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer n is the frequency span of its prime indices (row n of A296150).

Row-products of A325277.
The prime indices of a(n) are row n of A325757.
The unsorted prime signature of a(n) is row n of A325758.
Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A323014, A324843, A325248, A325249, A325759.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
Table[Times@@Prime/@freqspan[primeMS[n]],{n,30}]

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A325757 Irregular triangle read by rows giving the frequency span of n.

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A325759 Number of distinct frequencies in the frequency span of n.

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A325760 Heinz number of the frequency span of n.

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