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%I #9 May 15 2022 11:45:53
%S 2,4,8,9,15,16,18,21,27,30,32,33,36,39,42,45,51,54,57,60,63,64,66,69,
%T 72,78,81,84,87,90,93,99,102,108,111,114,117,120,123,125,126,128,129,
%U 132,135,138,141,144,153,156,159,162,168,171,174,175,177,180,183
%N Heinz numbers of integer partitions y with a fixed point y(i) = i. Such a fixed point is unique if it exists.
%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.
%e The terms together with their prime indices begin:
%e 2: (1)
%e 4: (1,1)
%e 8: (1,1,1)
%e 9: (2,2)
%e 15: (3,2)
%e 16: (1,1,1,1)
%e 18: (2,2,1)
%e 21: (4,2)
%e 27: (2,2,2)
%e 30: (3,2,1)
%e 32: (1,1,1,1,1)
%e 33: (5,2)
%e 36: (2,2,1,1)
%e 39: (6,2)
%e 42: (4,2,1)
%e 45: (3,2,2)
%e 51: (7,2)
%e 54: (2,2,2,1)
%e For example, the partition (3,2,2) with Heinz number 45 has a fixed point at position 2, so 45 is in the sequence.
%t pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t Select[Range[100],pq[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]==1&]
%Y * = unproved
%Y *These partitions are counted by A001522, strict A352829.
%Y *The complement is A352826, counted by A064428.
%Y The complement reverse version is A352830, counted by A238394.
%Y The reverse version is A352872, counted by A238395
%Y A000700 counts self-conjugate partitions, ranked by A088902.
%Y A001222 counts prime indices, distinct A001221.
%Y A008290 counts permutations by fixed points, unfixed A098825.
%Y A056239 adds up prime indices, row sums of A112798 and A296150.
%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 A238349 counts compositions by fixed points, complement A352523.
%Y A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A352828 counts strict partitions without a fixed point.
%Y A352833 counts partitions by fixed points.
%Y Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824 (characteristic function), A352825, A352831.
%K nonn
%O 1,1
%A _Gus Wiseman_, Apr 06 2022
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