cp's OEIS Frontend

A381871 Numbers whose prime indices cannot be partitioned into constant blocks having a common sum.

%I A381871 #9 Apr 27 2025 09:09:27
%S A381871 6,10,14,15,18,20,21,22,24,26,28,30,33,34,35,38,39,42,44,45,46,50,51,
%T A381871 52,54,55,56,57,58,60,62,65,66,68,69,70,72,74,75,76,77,78,80,82,84,85,
%U A381871 86,87,88,90,91,92,93,94,95,96,98,99,100,102,104,105,106,110
%N A381871 Numbers whose prime indices cannot be partitioned into constant blocks having a common sum.
%C A381871 First differs from A383100 in lacking 108.
%C A381871 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, sum A056239.
%C A381871 Also numbers that cannot be written as a product of prime powers with equal sums of prime indices.
%C A381871 Partitions of this type are counted by A381993.
%e A381871 The terms together with their prime indices begin:
%e A381871     6: {1,2}
%e A381871    10: {1,3}
%e A381871    14: {1,4}
%e A381871    15: {2,3}
%e A381871    18: {1,2,2}
%e A381871    20: {1,1,3}
%e A381871    21: {2,4}
%e A381871    22: {1,5}
%e A381871    24: {1,1,1,2}
%e A381871    26: {1,6}
%e A381871    28: {1,1,4}
%e A381871    30: {1,2,3}
%t A381871 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381871 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381871 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A381871 Select[Range[100],Select[mps[prix[#]],SameQ@@Total/@#&&And@@SameQ@@@#&]=={}&]
%Y A381871 Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).
%Y A381871 Constant blocks with distinct sums: A381635, A381716.
%Y A381871 For distinct instead of equal sums we have A381636, counted by A381717.
%Y A381871 Partitions of this type are counted by A381993, complement A383093.
%Y A381871 These are the positions of 0 in A381995.
%Y A381871 A001055 counts multiset partitions of prime indices, strict A045778.
%Y A381871 A050361 counts multiset partitions into distinct constant blocks.
%Y A381871 A055396 gives least prime index, greatest A061395.
%Y A381871 A056239 adds up prime indices, row sums of A112798.
%Y A381871 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A381871 Cf. A000720, A000961, A001222, A265947, A321469, A381633, A381715, A381719, A381806.
%K A381871 nonn
%O A381871 1,1
%A A381871 _Gus Wiseman_, Mar 13 2025