cp's OEIS Frontend

A384011 Numbers k such that it is not possible to choose disjoint strict integer partitions of each conjugate prime index of k.

%I A384011 #12 Jun 08 2025 14:20:12
%S A384011 3,5,7,9,10,11,13,14,15,17,19,20,21,22,23,25,26,28,29,31,33,34,35,37,
%T A384011 38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,55,56,57,58,59,61,62,63,
%U A384011 65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,82,83,84,85
%N A384011 Numbers k such that it is not possible to choose disjoint strict integer partitions of each conjugate prime index of k.
%C A384011 A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798.
%e A384011 The terms together with their prime indices begin:
%e A384011     3: {2}
%e A384011     5: {3}
%e A384011     7: {4}
%e A384011     9: {2,2}
%e A384011    10: {1,3}
%e A384011    11: {5}
%e A384011    13: {6}
%e A384011    14: {1,4}
%e A384011    15: {2,3}
%e A384011    17: {7}
%e A384011    19: {8}
%e A384011    20: {1,1,3}
%e A384011    21: {2,4}
%e A384011    22: {1,5}
%e A384011    23: {9}
%e A384011    25: {3,3}
%e A384011    26: {1,6}
%e A384011    28: {1,1,4}
%t A384011 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
%t A384011 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A384011 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384011 Select[Range[100],pof[conj[prix[#]]]=={}&]
%Y A384011 The conjugate is A382912.
%Y A384011 These complement is counted by A383708, ranks A382913 or A384010.
%Y A384011 These partitions are counted by A383710, conjugate A383711.
%Y A384011 These are the positions of 0 in A384005, conjugate A383706.
%Y A384011 A000041 counts integer partitions, strict A000009.
%Y A384011 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384011 A055396 gives least prime index, greatest A061395.
%Y A384011 A056239 adds up prime indices, row sums of A112798.
%Y A384011 A122111 represent conjugation in terms of Heinz numbers.
%Y A384011 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384011 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384011 Cf. A098859, A130091, A279375, A279790, A357982, A382525, A383533, A383707.
%K A384011 nonn
%O A384011 1,1
%A A384011 _Gus Wiseman_, May 23 2025