A352822 Number of fixed points y(i) = i, where y is the weakly increasing sequence of prime indices of n.

0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 1, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 0, 1, 0

Offset

1,6

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = A001222(n) - A352823(n). - Antti Karttunen, Apr 11 2022

Example

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

Maple

f:= proc(n) local F, J, t;
  F:= sort(ifactors(n)[2], (s, t) -> s[1]<t[1]);
  J:= map(t -> numtheory:-pi(t[1])$t[2], F);
  nops(select(t -> J[t]=t, [$1..nops(J)]));
end proc:
map(f, [$1..200]); # Robert Israel, Apr 11 2023

Mathematica

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

Prog

(PARI) A352822(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

Crossrefs

* = unproved

Positions of first appearances are A002110.

The triangle version is A238352.

Positions of 0's are A352830, counted by A238394.

Positions of 1's are A352831, counted by A352832.

A version for compositions is A352512, complement A352513, triangle A238349.

The complement is A352823.

The reverse version is A352824, complement A352825.

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.

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

Cf. A065770, A093641, A098825, A114088, A118199, A257990, A325163, A325164, A325169, A342192, A352486-A352491, A352829.

Sequence in context: A357924 A351563 A362983 * A090418 A363877 A322452

Adjacent sequences: A352819 A352820 A352821 * A352823 A352824 A352825

Keyword

nonn

Author

Gus Wiseman, Apr 05 2022

Extensions

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

Status

approved