OFFSET
1
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Table[pq[Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]], {n, 100}]
PROG
(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
CROSSREFS
* = unproved
The reverse version is A352822.
The reverse complement version is A352823.
The complement version is A352825.
A122111 represents partition conjugation using Heinz numbers.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 05 2022
EXTENSIONS
Data section extended up to 108 terms by Antti Karttunen, Apr 14 2022
STATUS
approved