A352825 Number of nonfixed points y(i) != i, where y is the integer partition with Heinz number n.

0, 0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 1, 5, 2, 3, 1, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 2, 5, 2, 2, 1, 3, 1, 3, 1, 4, 1, 2, 3, 3, 2, 2, 1, 5, 3, 2, 1, 3, 2, 2, 1, 4, 1, 3, 2, 3, 1, 2, 2, 6, 1, 3, 2, 4, 1, 2, 1, 4, 3

Offset: 1

Author

Gus Wiseman, Apr 05 2022

Keywords

• nonn

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.

Examples

			The partition (3,2,2,1) has Heinz number 90, so a(90) = 3. The partition (3,3,1,1) has Heinz number 100, so a(100) = 4.
		

Crossrefs

* = unproved

Positions of first appearances are A003945.

The version for standard compositions is A352513, complement A352512.

A corresponding triangle for compositions is A352523, complement A238349.

The reverse complement version is A352822, triangle A238352.

The reverse version is A352823.

The complement version is A352824, triangle version A352833.

A000700 counts self-conjugate partitions, ranked by A088902.

A001222 counts prime indices, distinct A001221.

*A001522 counts partitions with a fixed point, ranked by A352827.

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

*A064428 counts partitions without a fixed point, ranked by A352826.

A115720 and A115994 count partitions by their Durfee square.

A122111 represents partition conjugation using Heinz numbers.

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

A238394 counts reversed partitions without a fixed point, ranked by A352830.

A238395 counts reversed partitions with a fixed point, ranked by A352872.

A352832 counts reversed partitions with one fixed point, ranked by A352831.

Cf. A065770, A093641, A114088, A252464, A257990, A325163, A325164, A325165, A325169, A342192, A352486-A352491, A352828.

Programs

Mathematica

pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]];
Table[pnq[Reverse[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

PARI

A352825(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

Formula

a(n) = A001222(n) - A352824(n).

Extensions

Data section extended up to 105 terms by Antti Karttunen, Apr 14 2022