cp's OEIS Frontend

A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

%I A383014 #8 Apr 25 2025 08:46:52
%S A383014 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,25,27,29,31,32,36,37,40,41,43,
%T A383014 47,48,49,53,59,61,63,64,67,71,73,79,81,83,89,97,101,103,107,108,109,
%U A383014 112,113,121,125,127,128,131,137,139,144,149,151,157,163,167,169
%N A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.
%C A383014 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.
%e A383014 The prime indices of 36 are {1,1,2,2}, and a partition into constant blocks with a common sum is: {{2},{2},{1,1}}, so 36 is in the sequence.
%e A383014 The prime indices of 43200 are {1,1,1,1,1,1,2,2,2,3,3}, and a partition into constant blocks with a common sum is: {{{1,1,1,1,1,1},{2,2,2},{3,3}}}, so 43200 is in the sequence.
%e A383014 The prime indices of 520000 are {1,1,1,1,1,1,3,3,3,3,6} and a partition into constant blocks with a common sum is: {{1,1,1,1,1,1},{3,3},{3,3},{6}}, so 520000 is in the sequence.
%e A383014 The terms together with their prime indices begin:
%e A383014    1: {}
%e A383014    2: {1}
%e A383014    3: {2}
%e A383014    4: {1,1}
%e A383014    5: {3}
%e A383014    7: {4}
%e A383014    8: {1,1,1}
%e A383014    9: {2,2}
%e A383014   11: {5}
%e A383014   12: {1,1,2}
%e A383014   13: {6}
%e A383014   16: {1,1,1,1}
%e A383014   17: {7}
%e A383014   19: {8}
%e A383014   23: {9}
%e A383014   25: {3,3}
%e A383014   27: {2,2,2}
%e A383014   29: {10}
%e A383014   31: {11}
%e A383014   32: {1,1,1,1,1}
%e A383014   36: {1,1,2,2}
%e A383014   37: {12}
%e A383014   40: {1,1,1,3}
%t A383014 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383014 mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn, {ptn,IntegerPartitions[Length[y]]}];
%t A383014 Select[Range[100], Select[Join@@@Tuples[mce/@Split[prix[#]]], SameQ@@Total/@#&]!={}&]
%Y A383014 Twice-partitions of this type (constant blocks with a common sum) are counted by A279789.
%Y A383014 Includes all elements of A353833.
%Y A383014 For distinct sums we have the complement of A381636.
%Y A383014 For strict blocks we have the complement of A381719.
%Y A383014 For distinct sums and strict blocks we have the complement of A381806.
%Y A383014 The complement is A381871, counted by A381993.
%Y A383014 These are the positions of positive terms in A381995.
%Y A383014 Partitions of this type are counted by A383093.
%Y A383014 Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).
%Y A383014 A001055 counts factorizations (multiset partitions of prime indices), strict A045778.
%Y A383014 A050361 counts factorizations into distinct prime powers.
%Y A383014 A055396 gives least prime index, greatest A061395.
%Y A383014 A056239 adds up prime indices, row sums of A112798.
%Y A383014 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A383014 Cf. A000720, A000961, A001222, A321469, A381635, A381715, A381716, A381717.
%K A383014 nonn
%O A383014 1,2
%A A383014 _Gus Wiseman_, Apr 22 2025