A352824 Number of fixed points y(i) = i, where y is the integer partition with Heinz number n.

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

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.

Links

Table of n, a(n) for n=1..108.

Index entries for characteristic functions

Formula

a(n) = A001222(n) - A352825(n).

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

A version for standard compositions is A352512, complement A352513.

The reverse version is A352822.

The reverse complement version is A352823.

The complement version is A352825.

Positions of 0's are A352826, counted by *A064428.

Characteristic function of A352827 (positions of 1's), counted by *A001522.

The corresponding triangle (zeros removed) is A352833, reverse A238352.

A352832 counts partitions with a fixed point, ranked by A352831.

A000700 counts self-conjugate partitions, ranked by A088902.

A001222 counts prime indices, distinct A001221.

A056239 adds up prime indices, row sums of A112798 and A296150.

A115720 and A115994 count partitions by Durfee square.

A122111 represents partition conjugation using Heinz numbers.

A124010 gives prime signature, sorted A118914, conjugate rank A238745.

A238349 counts compositions by fixed points, complement A352523.

A238394 counts partitions without fixed points, ranked by A352830.

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.

Sequence in context: A289741 A127266 A083923 * A101309 A141474 A073424

Adjacent sequences: A352821 A352822 A352823 * A352825 A352826 A352827

Keyword

nonn

Author

Gus Wiseman, Apr 05 2022

Extensions

Data section extended up to 108 terms by Antti Karttunen, Apr 14 2022

Status

approved