A321453 - OEIS

%I #6 Nov 13 2018 12:54:32

%S 1,2,3,5,6,7,10,11,13,14,15,17,18,19,20,21,22,23,24,26,28,29,31,33,34,

%T 35,37,38,39,41,42,43,44,45,46,47,50,51,52,53,54,55,56,57,58,59,60,61,

%U 62,65,66,67,68,69,71,72,73,74,75,76,77,78,79,80,82,83,85

%N Numbers that cannot be factored into two or more factors all having the same sum of prime indices.

%C Also Heinz numbers of integer partitions that cannot be partitioned into two or more blocks with equal sums. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%C 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. The sum of prime indices of n is A056239(n).

%e The sequence of all integer partitions that cannot be partitioned into two or more blocks with equal sums begins: (1), (2), (3), (21), (4), (31), (5), (6), (41), (32), (7), (221), (8), (311), (42), (51), (9), (2111), (61), (411).

%t hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Select[Range[100],Select[facs[#],And[Length[#]>1,SameQ@@hwt/@#]&]=={}&]

%Y Positions of 1's in A321455.

%Y Cf. A056239, A112798, A276024, A279787, A305551, A306017, A317144, A320322, A321451, A321452, A321454.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 10 2018