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A352827 Heinz numbers of integer partitions y with a fixed point y(i) = i. Such a fixed point is unique if it exists.

%I A352827 #9 May 15 2022 11:45:53
%S A352827 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 A352827 72,78,81,84,87,90,93,99,102,108,111,114,117,120,123,125,126,128,129,
%U A352827 132,135,138,141,144,153,156,159,162,168,171,174,175,177,180,183
%N A352827 Heinz numbers of integer partitions y with a fixed point y(i) = i. Such a fixed point is unique if it exists.
%C A352827 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 A352827 The terms together with their prime indices begin:
%e A352827     2: (1)
%e A352827     4: (1,1)
%e A352827     8: (1,1,1)
%e A352827     9: (2,2)
%e A352827    15: (3,2)
%e A352827    16: (1,1,1,1)
%e A352827    18: (2,2,1)
%e A352827    21: (4,2)
%e A352827    27: (2,2,2)
%e A352827    30: (3,2,1)
%e A352827    32: (1,1,1,1,1)
%e A352827    33: (5,2)
%e A352827    36: (2,2,1,1)
%e A352827    39: (6,2)
%e A352827    42: (4,2,1)
%e A352827    45: (3,2,2)
%e A352827    51: (7,2)
%e A352827    54: (2,2,2,1)
%e A352827 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 A352827 pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t A352827 Select[Range[100],pq[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]==1&]
%Y A352827 * = unproved
%Y A352827 *These partitions are counted by A001522, strict A352829.
%Y A352827 *The complement is A352826, counted by A064428.
%Y A352827 The complement reverse version is A352830, counted by A238394.
%Y A352827 The reverse version is A352872, counted by A238395
%Y A352827 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A352827 A001222 counts prime indices, distinct A001221.
%Y A352827 A008290 counts permutations by fixed points, unfixed A098825.
%Y A352827 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A352827 A115720 and A115994 count partitions by their Durfee square.
%Y A352827 A122111 represents partition conjugation using Heinz numbers.
%Y A352827 A124010 gives prime signature, sorted A118914, conjugate rank A238745.
%Y A352827 A238349 counts compositions by fixed points, complement A352523.
%Y A352827 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A352827 A352828 counts strict partitions without a fixed point.
%Y A352827 A352833 counts partitions by fixed points.
%Y A352827 Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824 (characteristic function), A352825, A352831.
%K A352827 nonn
%O A352827 1,1
%A A352827 _Gus Wiseman_, Apr 06 2022