A371955 Numbers with triquanimous prime indices.

8, 27, 36, 48, 64, 125, 150, 180, 200, 216, 240, 288, 320, 343, 384, 441, 490, 512, 567, 588, 630, 700, 729, 756, 784, 810, 840, 900, 972, 1000, 1008, 1080, 1120, 1200, 1296, 1331, 1344, 1440, 1600, 1694, 1728, 1792, 1815, 1920, 2156, 2178, 2197, 2304, 2310

Offset: 1

Author

Gus Wiseman, Apr 19 2024

Keywords

• nonn

Comments

A finite multiset of numbers is defined to be triquanimous iff it can be partitioned into three multisets with equal sums.

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.

Examples

			The terms together with their prime indices begin:
     8: {1,1,1}
    27: {2,2,2}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    64: {1,1,1,1,1,1}
   125: {3,3,3}
   150: {1,2,3,3}
   180: {1,1,2,2,3}
   200: {1,1,1,3,3}
   216: {1,1,1,2,2,2}
   240: {1,1,1,1,2,3}
   288: {1,1,1,1,1,2,2}
   320: {1,1,1,1,1,1,3}
   343: {4,4,4}
   384: {1,1,1,1,1,1,1,2}
   441: {2,2,4,4}
   490: {1,3,4,4}
   512: {1,1,1,1,1,1,1,1,1}
   567: {2,2,2,2,4}
   588: {1,1,2,4,4}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..500

Crossrefs

These are the Heinz numbers of the partitions counted by A002220.

For biquanimous we have A357976, counted by A002219.

For non-biquanimous we have A371731, counted by A371795, even case A006827.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A237258 (aerated) counts biquanimous strict partitions, ranks A357854.

A371783 counts k-quanimous partitions.

Cf. A000005, A002221, A035470, A064914, A232466, A299701, A321142, A321454, A321455, A326534, A357879, A371781.

Programs

Maple

tripart:= proc(L) local t,X,Y,n,cons,i,R;
  t:= convert(L,`+`)/3;
  n:= nops(L);
  if not t::integer then return false fi;
  cons:= [add(L[i]*X[i],i=1..n)=t,
          add(L[i]*Y[i],i=1..n)=t,
          seq(X[i] + Y[i] <= 1, i=1..n)];
  R:= traperror(Optimization:-Maximize(0, cons, assume=binary));
  R::list
end proc:
primeindices:= proc(n) local F,t;
  F:= ifactors(n)[2];
  map(t -> numtheory:-pi(t[1])$t[2], F)
end proc:
select(tripart @ primindices, [$2..3000]); # Robert Israel, May 19 2025

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[1000],Select[facs[#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]