cp's OEIS Frontend

A384321 Numbers whose distinct prime indices are not maximally refined.

%I A384321 #10 Jun 11 2025 23:40:56
%S A384321 5,7,11,13,17,19,21,22,23,25,26,29,31,33,34,35,37,38,39,41,43,46,47,
%T A384321 49,51,53,55,57,58,59,61,62,65,67,69,71,73,74,77,79,82,83,85,86,87,89,
%U A384321 91,93,94,95,97,101,102,103,106,107,109,111,113,114,115,118,119
%N A384321 Numbers whose distinct prime indices are not maximally refined.
%C A384321 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A384321 Given a partition, the following are equivalent:
%C A384321 1) The distinct parts are maximally refined.
%C A384321 2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
%C A384321 3) No part is a sum of distinct non-parts.
%e A384321 The prime indices of 25 are {3,3}, which has refinements: ((3),(1,2)) and ((1,2),(3)), so 25 is in the sequence.
%e A384321 The prime indices of 102 are {1,2,7}, which has refinement ((1),(2),(3,4)), so 102 is in the sequence.
%e A384321 The terms together with their prime indices begin:
%e A384321      5: {3}      39: {2,6}      73: {21}
%e A384321      7: {4}      41: {13}       74: {1,12}
%e A384321     11: {5}      43: {14}       77: {4,5}
%e A384321     13: {6}      46: {1,9}      79: {22}
%e A384321     17: {7}      47: {15}       82: {1,13}
%e A384321     19: {8}      49: {4,4}      83: {23}
%e A384321     21: {2,4}    51: {2,7}      85: {3,7}
%e A384321     22: {1,5}    53: {16}       86: {1,14}
%e A384321     23: {9}      55: {3,5}      87: {2,10}
%e A384321     25: {3,3}    57: {2,8}      89: {24}
%e A384321     26: {1,6}    58: {1,10}     91: {4,6}
%e A384321     29: {10}     59: {17}       93: {2,11}
%e A384321     31: {11}     61: {18}       94: {1,15}
%e A384321     33: {2,5}    62: {1,11}     95: {3,8}
%e A384321     34: {1,7}    65: {3,6}      97: {25}
%e A384321     35: {3,4}    67: {19}      101: {26}
%e A384321     37: {12}     69: {2,9}     102: {1,2,7}
%e A384321     38: {1,8}    71: {20}      103: {27}
%t A384321 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384321 nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
%t A384321 Select[Range[30],With[{y=Union[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]!={}]&]
%Y A384321 These appear to be positions of terms > 1 in A383706, non-disjoint A357982, non-strict A299200.
%Y A384321 The strict complement is A383707, counted by A179009.
%Y A384321 Partitions of this type appear to be counted by A384317.
%Y A384321 The complement is A384320.
%Y A384321 The strict (squarefree) case appears to be A384322, counted by A384318.
%Y A384321 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384321 A055396 gives least prime index, greatest A061395.
%Y A384321 A056239 adds up prime indices, row sums of A112798.
%Y A384321 A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
%Y A384321 A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
%Y A384321 A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
%Y A384321 Cf. A098859, A122111, A130091, A317142, A326080, A381454, A382525, A384005, A384323, A384390.
%K A384321 nonn
%O A384321 1,1
%A A384321 _Gus Wiseman_, Jun 01 2025