A352823 Number of nonfixed points y(i) != i, where y is the weakly increasing sequence of prime indices of n.

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

Offset: 1

Author

Gus Wiseman, Apr 05 2022

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The prime indices of 6500 are {1,1,3,3,3,6}, with nonfixed points at positions {2,4,5}, so a(6500) = 3.
		

Links

• Index entries for sequences computed from indices in prime factorization

Crossrefs

* = unproved

Positions of zeros are A002110

Positions of first appearances are A077552.

The complement triangle version is A238352.

A version for compositions is A352513, complement A352512.

The complement is A352822.

The reverse version is A352825, complement A352824.

Complement positions of 1's are A352831, counted by A352832.

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.

A122111 represents partition conjugation using Heinz numbers.

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

A115720 and A115994 count partitions by their Durfee square.

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

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

A352833 counts partitions by fixed points, rank statistic A352824.

Cf. A065770, A093641, A114088, A252464, A257990, A325163, A325164, A325165, A325169, A342192, A352486-A352491, A352828, A352829.

Programs

Mathematica

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

PARI

A352823(n) = { my(f=factor(n),i=0,c=0); for(k=1,#f~,while(f[k,2], f[k,2]--; i++; c += (i!=primepi(f[k,1])))); (c); }; \\ Antti Karttunen, Apr 11 2022

Formula

a(n) = A001222(n) - A352822(n). - Antti Karttunen, Apr 11 2022

Extensions

Data section extended up to 105 terms by Antti Karttunen, Apr 11 2022