cp's OEIS Frontend

A381636 Numbers whose prime indices cannot be partitioned into constant blocks with distinct sums.

%I A381636 #11 Apr 26 2025 23:33:00
%S A381636 12,60,63,84,120,126,132,156,204,228,252,276,300,315,325,348,372,420,
%T A381636 444,492,504,516,560,564,588,630,636,650,660,693,708,720,732,780,804,
%U A381636 819,840,852,876,924,931,948,975,996,1008,1020,1068,1071,1092,1140,1164
%N A381636 Numbers whose prime indices cannot be partitioned into constant blocks with distinct sums.
%C A381636 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 A381636 Also numbers that cannot be written as a product of prime powers > 1 with distinct sums of prime indices (A056239).
%C A381636 Contains no squarefree numbers.
%C A381636 Conjecture: These are the zeros of A382876.
%e A381636 The prime indices of 300 are {1,1,2,3,3}, with partitions into constant blocks:
%e A381636   {{2},{1,1},{3,3}}
%e A381636   {{1},{1},{2},{3,3}}
%e A381636   {{2},{3},{3},{1,1}}
%e A381636   {{1},{1},{2},{3},{3}}
%e A381636 but none of these has distinct block-sums, so 300 is in the sequence.
%e A381636 The terms together with their prime indices begin:
%e A381636    12: {1,1,2}
%e A381636    60: {1,1,2,3}
%e A381636    63: {2,2,4}
%e A381636    84: {1,1,2,4}
%e A381636   120: {1,1,1,2,3}
%e A381636   126: {1,2,2,4}
%e A381636   132: {1,1,2,5}
%e A381636   156: {1,1,2,6}
%e A381636   204: {1,1,2,7}
%e A381636   228: {1,1,2,8}
%e A381636   252: {1,1,2,2,4}
%e A381636   276: {1,1,2,9}
%e A381636   300: {1,1,2,3,3}
%t A381636 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A381636 pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
%t A381636 Select[Range[100],Select[pfacs[#],UnsameQ@@hwt/@#&]=={}&]
%Y A381636 More on multiset partitions into constant blocks: A006171, A279784, A295935.
%Y A381636 These are the positions of 0 in A381635, after taking block-sums A381716.
%Y A381636 Partitions of this type are counted by A381717.
%Y A381636 For strict instead of constant blocks we have A381806, zeros of A381633.
%Y A381636 For equal instead of distinct block-sums we have A381871.
%Y A381636 A000688 counts multiset partitions into constant, see A381455 (upper), A381453 (lower).
%Y A381636 A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
%Y A381636 A050361 counts multiset partitions into distinct constant blocks, after sums A381715.
%Y A381636 A055396 gives least prime index, greatest A061395.
%Y A381636 A056239 adds up prime indices, row sums of A112798.
%Y A381636 Cf. A000720, A001222, A005117, A050320, A059404, A213242, A293243, A299202, A300385, A381078, A381454, A381634.
%K A381636 nonn
%O A381636 1,1
%A A381636 _Gus Wiseman_, Mar 10 2025