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A352826 Heinz numbers of integer partitions y without a fixed point y(i) = i. Such a fixed point is unique if it exists.

%I A352826 #9 May 15 2022 11:45:24
%S A352826 1,3,5,6,7,10,11,12,13,14,17,19,20,22,23,24,25,26,28,29,31,34,35,37,
%T A352826 38,40,41,43,44,46,47,48,49,50,52,53,55,56,58,59,61,62,65,67,68,70,71,
%U A352826 73,74,75,76,77,79,80,82,83,85,86,88,89,91,92,94,95,96,97
%N A352826 Heinz numbers of integer partitions y without a fixed point y(i) = i. Such a fixed point is unique if it exists.
%C A352826 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 A352826 The terms together with their prime indices begin:
%e A352826       1: ()          24: (2,1,1,1)     47: (15)
%e A352826       3: (2)         25: (3,3)         48: (2,1,1,1,1)
%e A352826       5: (3)         26: (6,1)         49: (4,4)
%e A352826       6: (2,1)       28: (4,1,1)       50: (3,3,1)
%e A352826       7: (4)         29: (10)          52: (6,1,1)
%e A352826      10: (3,1)       31: (11)          53: (16)
%e A352826      11: (5)         34: (7,1)         55: (5,3)
%e A352826      12: (2,1,1)     35: (4,3)         56: (4,1,1,1)
%e A352826      13: (6)         37: (12)          58: (10,1)
%e A352826      14: (4,1)       38: (8,1)         59: (17)
%e A352826      17: (7)         40: (3,1,1,1)     61: (18)
%e A352826      19: (8)         41: (13)          62: (11,1)
%e A352826      20: (3,1,1)     43: (14)          65: (6,3)
%e A352826      22: (5,1)       44: (5,1,1)       67: (19)
%e A352826      23: (9)         46: (9,1)         68: (7,1,1)
%t A352826 pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t A352826 Select[Range[100],pq[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]==0&]
%Y A352826 * = unproved
%Y A352826 *These partitions are counted by A064428, strict A352828.
%Y A352826 The complement is A352827.
%Y A352826 The reverse version is A352830, counted by A238394.
%Y A352826 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A352826 A001222 counts prime indices, distinct A001221.
%Y A352826 *A001522 counts partitions with a fixed point.
%Y A352826 A008290 counts permutations by fixed points, nonfixed A098825.
%Y A352826 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A352826 A115720 and A115994 count partitions by their Durfee square.
%Y A352826 A122111 represents partition conjugation using Heinz numbers.
%Y A352826 A124010 gives prime signature, sorted A118914.
%Y A352826 A238349 counts compositions by fixed points, complement A352523.
%Y A352826 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A352826 A238395 counts reversed partitions with a fixed point, ranked by A352872.
%Y A352826 A352833 counts partitions by fixed points.
%Y A352826 Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824, A352825, A352831, A352832.
%K A352826 nonn
%O A352826 1,2
%A A352826 _Gus Wiseman_, Apr 06 2022