cp's OEIS Frontend

A321454 Numbers that can be factored into two or more factors all having the same sum of prime indices.

%I A321454 #6 Nov 13 2018 12:54:41
%S A321454 4,8,9,12,16,25,27,30,32,36,40,48,49,63,64,70,81,84,90,100,108,112,
%T A321454 120,121,125,128,144,150,154,160,165,169,180,192,196,198,200,210,216,
%U A321454 220,225,240,243,252,256,264,270,273,280,286,288,289,300,320,324,325
%N A321454 Numbers that can be factored into two or more factors all having the same sum of prime indices.
%C A321454 Also Heinz numbers of integer partitions that can 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 A321454 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 A321454 The sequence of all integer partitions that can be partitioned into two or more blocks with equal sums begins: (11), (111), (22), (211), (1111), (33), (222), (321), (11111), (2211), (3111), (21111), (44), (422), (111111), (431), (2222), (4211), (3221), (3311), (22211), (41111), (32111), (55), (333), (1111111), (221111), (3321), (541), (311111), (532), (66), (32211), (2111111), (4411), (5221), (33111).
%e A321454 The Heinz number of (32111) is 120, which has factorization (10*12) corresponding to the multiset partition ((13)(112)) whose blocks have equal sums, so 120 belongs to the sequence.
%t A321454 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A321454 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A321454 Select[Range[100],Select[facs[#],And[Length[#]>1,SameQ@@hwt/@#]&]!={}&]
%Y A321454 Positions of terms > 1 in A321455.
%Y A321454 Cf. A056239, A112798, A276024, A279787, A305551, A306017, A317144, A320322, A321451, A321452, A321453.
%K A321454 nonn
%O A321454 1,1
%A A321454 _Gus Wiseman_, Nov 10 2018