A352825 - OEIS

%I #13 May 25 2022 06:43:11

%S 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,

%T 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,

%U 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

%N Number of nonfixed 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.

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

%e 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.

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

%t Table[pnq[Reverse[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

%o (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

%Y * = unproved

%Y Positions of first appearances are A003945.

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

%Y A corresponding triangle for compositions is A352523, complement A238349.

%Y The reverse complement version is A352822, triangle A238352.

%Y The reverse version is A352823.

%Y The complement version is A352824, triangle version A352833.

%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 A115720 and A115994 count partitions by their Durfee square.

%Y A122111 represents partition conjugation using Heinz numbers.

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

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

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

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

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

%K nonn

%O 1,6

%A _Gus Wiseman_, Apr 05 2022

%E Data section extended up to 105 terms by _Antti Karttunen_, Apr 14 2022