A321453 Numbers that cannot be factored into two or more factors all having the same sum of prime indices.

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, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 85

Offset: 1

Author

Gus Wiseman, Nov 10 2018

Keywords

• nonn

Comments

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).

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).

Examples

			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).
		

Crossrefs

Positions of 1's in A321455.

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

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],And[Length[#]>1,SameQ@@hwt/@#]&]=={}&]