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

4, 8, 9, 12, 16, 25, 27, 30, 32, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 125, 128, 144, 150, 154, 160, 165, 169, 180, 192, 196, 198, 200, 210, 216, 220, 225, 240, 243, 252, 256, 264, 270, 273, 280, 286, 288, 289, 300, 320, 324, 325

Offset

1,1

Comments

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).

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).

Links

Table of n, a(n) for n=1..56.

Example

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).
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.

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/@#]&]!={}&]

Crossrefs

Positions of terms > 1 in A321455.

Cf. A056239, A112798, A276024, A279787, A305551, A306017, A317144, A320322, A321451, A321452, A321453.

Sequence in context: A374037 A102211 A244032 * A353834 A319302 A119025

Adjacent sequences: A321451 A321452 A321453 * A321455 A321456 A321457

Keyword

nonn

Author

Gus Wiseman, Nov 10 2018

Status

approved