# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a352825 Showing 1-1 of 1 %I A352825 #13 May 25 2022 06:43:11 %S A352825 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 A352825 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 A352825 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 A352825 Number of nonfixed points y(i) != i, where y is the integer partition with Heinz number n. %C A352825 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 A352825 a(n) = A001222(n) - A352824(n). %e A352825 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 A352825 pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]]; %t A352825 Table[pnq[Reverse[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}] %o A352825 (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 A352825 * = unproved %Y A352825 Positions of first appearances are A003945. %Y A352825 The version for standard compositions is A352513, complement A352512. %Y A352825 A corresponding triangle for compositions is A352523, complement A238349. %Y A352825 The reverse complement version is A352822, triangle A238352. %Y A352825 The reverse version is A352823. %Y A352825 The complement version is A352824, triangle version A352833. %Y A352825 A000700 counts self-conjugate partitions, ranked by A088902. %Y A352825 A001222 counts prime indices, distinct A001221. %Y A352825 *A001522 counts partitions with a fixed point, ranked by A352827. %Y A352825 A056239 adds up prime indices, row sums of A112798 and A296150. %Y A352825 *A064428 counts partitions without a fixed point, ranked by A352826. %Y A352825 A115720 and A115994 count partitions by their Durfee square. %Y A352825 A122111 represents partition conjugation using Heinz numbers. %Y A352825 A124010 gives prime signature, sorted A118914, conjugate rank A238745. %Y A352825 A238394 counts reversed partitions without a fixed point, ranked by A352830. %Y A352825 A238395 counts reversed partitions with a fixed point, ranked by A352872. %Y A352825 A352832 counts reversed partitions with one fixed point, ranked by A352831. %Y A352825 Cf. A065770, A093641, A114088, A252464, A257990, A325163, A325164, A325165, A325169, A342192, A352486-A352491, A352828. %K A352825 nonn %O A352825 1,6 %A A352825 _Gus Wiseman_, Apr 05 2022 %E A352825 Data section extended up to 105 terms by _Antti Karttunen_, Apr 14 2022 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE