A325757 - OEIS

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

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OFFSET

1,2

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

Triangle begins:

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

MATHEMATICA

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}]

CROSSREFS

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.

Sequence in context: A246028 A232186 A340061 * A161161 A350357 A136277

Adjacent sequences: A325754 A325755 A325756 * A325758 A325759 A325760

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, May 19 2019

STATUS

approved