cp's OEIS Frontend

A352486 Heinz numbers of non-self-conjugate integer partitions.

%I A352486 #11 Mar 31 2022 03:03:53
%S A352486 3,4,5,7,8,10,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,
%T A352486 31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A352486 54,55,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73
%N A352486 Heinz numbers of non-self-conjugate integer partitions.
%C A352486 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. The sequence lists all Heinz numbers of partitions whose Heinz number is different from that of their conjugate.
%F A352486 a(n) != A122111(a(n)).
%e A352486 The terms together with their prime indices begin:
%e A352486    3: (2)
%e A352486    4: (1,1)
%e A352486    5: (3)
%e A352486    7: (4)
%e A352486    8: (1,1,1)
%e A352486   10: (3,1)
%e A352486   11: (5)
%e A352486   12: (2,1,1)
%e A352486   13: (6)
%e A352486   14: (4,1)
%e A352486   15: (3,2)
%e A352486   16: (1,1,1,1)
%e A352486   17: (7)
%e A352486   18: (2,2,1)
%e A352486 For example, the self-conjugate partition (4,3,3,1) has Heinz number 350, so 350 is not in the sequence.
%t A352486 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A352486 conj[y_]:=If[Length[y0]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A352486 Select[Range[100],#!=Times@@Prime/@conj[primeMS[#]]&]
%Y A352486 The complement is A088902, counted by A000700.
%Y A352486 These partitions are counted by A330644.
%Y A352486 These are the positions of nonzero terms in A352491.
%Y A352486 A000041 counts integer partitions, strict A000009.
%Y A352486 A098825 counts permutations by unfixed points.
%Y A352486 A238349 counts compositions by fixed points, rank statistic A352512.
%Y A352486 A325039 counts partitions w/ same product as conjugate, ranked by A325040.
%Y A352486 A352523 counts compositions by unfixed points, rank statistic A352513.
%Y A352486 Heinz number (rank) and partition:
%Y A352486 - A003963 = product of partition, conjugate A329382
%Y A352486 - A008480 = number of permutations of partition, conjugate A321648.
%Y A352486 - A056239 = sum of partition
%Y A352486 - A122111 = rank of conjugate partition
%Y A352486 - A296150 = parts of partition, reverse A112798, conjugate A321649
%Y A352486 - A352487 = less than conjugate, counted by A000701
%Y A352486 - A352488 = greater than or equal to conjugate, counted by A046682
%Y A352486 - A352489 = less than or equal to conjugate, counted by A046682
%Y A352486 - A352490 = greater than conjugate, counted by A000701
%Y A352486 Cf. A000720, A026424, A120383, A175508, A195017, A238745, A301987, A304360, A316524,  A324846, A350841.
%K A352486 nonn
%O A352486 1,1
%A A352486 _Gus Wiseman_, Mar 20 2022