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A325757
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Irregular triangle read by rows giving the frequency span of n.
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3
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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
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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}]
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CROSSREFS
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Number of distinct terms in row n is A325759(n).
Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A290822, A323014, A324843, A325277, A325755, A325760.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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