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A352824
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Number of fixed points y(i) = i, where y is the integer partition with Heinz number n.
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12
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0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1
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OFFSET
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1
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COMMENTS
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The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Conjecture: The mean approaches 1/4.
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LINKS
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FORMULA
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EXAMPLE
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The partition (3,2,2,1) has Heinz number 90, and when indexing the parts from left to right as k=1..4, then at k=2 we have y(k) = k [and nowhere else], therefore a(90) = 1. The partition (3,3,1,1) has Heinz number 100, with parts y(1) = 3, y(2) = 3, y(3) = 1, y(4) = 1, and as there are no such k that y(k) = k, we have a(100) = 0.
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MATHEMATICA
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pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Table[pq[Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]], {n, 100}]
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PROG
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(PARI) A352824(n) = { my(f=factor(n), i=bigomega(n), c=0); for(k=1, #f~, while(f[k, 2], f[k, 2]--; c += (i==primepi(f[k, 1])); i--)); (c); }; \\ Antti Karttunen, Apr 14 2022
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CROSSREFS
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* = unproved
The reverse complement version is A352823.
Characteristic function of A352827 (positions of 1's), counted by *A001522.
The corresponding triangle (zeros removed) is A352833, reverse A238352.
A122111 represents partition conjugation using Heinz numbers.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
Cf. A065770, A093641, A114088, A252464, A257990, A325163, A325164, A325165, A325169, A342192, A352486-A352491, A352829.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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