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%I #22 Apr 11 2023 12:06:24
%S 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,
%T 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,
%U 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
%N Number of fixed points y(i) = i, where y is the weakly increasing sequence of prime indices of n.
%C 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.
%H Robert Israel, <a href="/A352822/b352822.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F a(n) = A001222(n) - A352823(n). - _Antti Karttunen_, Apr 11 2022
%e The prime indices of 6500 are {1,1,3,3,3,6} with fixed points at positions {1,3,6}, so a(6500) = 3.
%p f:= proc(n) local F,J,t;
%p F:= sort(ifactors(n)[2],(s,t) -> s[1]<t[1]);
%p J:= map(t -> numtheory:-pi(t[1])$t[2], F);
%p nops(select(t -> J[t]=t, [$1..nops(J)]));
%p end proc:
%p map(f, [$1..200]); # _Robert Israel_, Apr 11 2023
%t pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t Table[pq[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]],{n,100}]
%o (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
%Y * = unproved
%Y Positions of first appearances are A002110.
%Y The triangle version is A238352.
%Y Positions of 0's are A352830, counted by A238394.
%Y Positions of 1's are A352831, counted by A352832.
%Y A version for compositions is A352512, complement A352513, triangle A238349.
%Y The complement is A352823.
%Y The reverse version is A352824, complement A352825.
%Y A000700 counts self-conjugate partitions, ranked by A088902.
%Y A001222 counts prime indices, distinct A001221.
%Y *A001522 counts partitions with a fixed point, ranked by A352827.
%Y A056239 adds up prime indices, row sums of A112798 and A296150.
%Y *A064428 counts partitions without a fixed point, ranked by A352826.
%Y A122111 represents partition conjugation using Heinz numbers.
%Y A124010 gives prime signature, sorted A118914, conjugate rank A238745.
%Y A115720 and A115994 count partitions by their Durfee square.
%Y A238395 counts reversed partitions with a fixed point, ranked by A352872.
%Y Cf. A065770, A093641, A098825, A114088, A118199, A257990, A325163, A325164, A325169, A342192, A352486-A352491, A352829.
%K nonn
%O 1,6
%A _Gus Wiseman_, Apr 05 2022
%E Data section extended up to 105 terms by _Antti Karttunen_, Apr 11 2022
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