A352824 - OEIS

%I #19 May 15 2022 11:46:59

%S 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,

%T 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,

%U 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

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

%C 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.

%C Conjecture: The mean approaches 1/4.

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

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

%e 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.

%t pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];

%t Table[pq[Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}]

%o (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

%Y * = unproved

%Y A version for standard compositions is A352512, complement A352513.

%Y The reverse version is A352822.

%Y The reverse complement version is A352823.

%Y The complement version is A352825.

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

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

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

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

%Y A000700 counts self-conjugate partitions, ranked by A088902.

%Y A001222 counts prime indices, distinct A001221.

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

%Y A115720 and A115994 count partitions by Durfee square.

%Y A122111 represents partition conjugation using Heinz numbers.

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

%Y A238349 counts compositions by fixed points, complement A352523.

%Y A238394 counts partitions without fixed points, ranked by A352830.

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

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

%K nonn

%O 1

%A _Gus Wiseman_, Apr 05 2022

%E Data section extended up to 108 terms by _Antti Karttunen_, Apr 14 2022