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A371955 Numbers with triquanimous prime indices.

%I A371955 #11 May 19 2025 16:33:09
%S A371955 8,27,36,48,64,125,150,180,200,216,240,288,320,343,384,441,490,512,
%T A371955 567,588,630,700,729,756,784,810,840,900,972,1000,1008,1080,1120,1200,
%U A371955 1296,1331,1344,1440,1600,1694,1728,1792,1815,1920,2156,2178,2197,2304,2310
%N A371955 Numbers with triquanimous prime indices.
%C A371955 A finite multiset of numbers is defined to be triquanimous iff it can be partitioned into three multisets with equal sums.
%C A371955 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.
%H A371955 Robert Israel, <a href="/A371955/b371955.txt">Table of n, a(n) for n = 1..500</a>
%e A371955 The terms together with their prime indices begin:
%e A371955      8: {1,1,1}
%e A371955     27: {2,2,2}
%e A371955     36: {1,1,2,2}
%e A371955     48: {1,1,1,1,2}
%e A371955     64: {1,1,1,1,1,1}
%e A371955    125: {3,3,3}
%e A371955    150: {1,2,3,3}
%e A371955    180: {1,1,2,2,3}
%e A371955    200: {1,1,1,3,3}
%e A371955    216: {1,1,1,2,2,2}
%e A371955    240: {1,1,1,1,2,3}
%e A371955    288: {1,1,1,1,1,2,2}
%e A371955    320: {1,1,1,1,1,1,3}
%e A371955    343: {4,4,4}
%e A371955    384: {1,1,1,1,1,1,1,2}
%e A371955    441: {2,2,4,4}
%e A371955    490: {1,3,4,4}
%e A371955    512: {1,1,1,1,1,1,1,1,1}
%e A371955    567: {2,2,2,2,4}
%e A371955    588: {1,1,2,4,4}
%p A371955 tripart:= proc(L) local t,X,Y,n,cons,i,R;
%p A371955   t:= convert(L,`+`)/3;
%p A371955   n:= nops(L);
%p A371955   if not t::integer then return false fi;
%p A371955   cons:= [add(L[i]*X[i],i=1..n)=t,
%p A371955           add(L[i]*Y[i],i=1..n)=t,
%p A371955           seq(X[i] + Y[i] <= 1, i=1..n)];
%p A371955   R:= traperror(Optimization:-Maximize(0, cons, assume=binary));
%p A371955   R::list
%p A371955 end proc:
%p A371955 primeindices:= proc(n) local F,t;
%p A371955   F:= ifactors(n)[2];
%p A371955   map(t -> numtheory:-pi(t[1])$t[2], F)
%p A371955 end proc:
%p A371955 select(tripart @ primindices, [$2..3000]); # _Robert Israel_, May 19 2025
%t A371955 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A371955 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A371955 Select[Range[1000],Select[facs[#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]
%Y A371955 These are the Heinz numbers of the partitions counted by A002220.
%Y A371955 For biquanimous we have A357976, counted by A002219.
%Y A371955 For non-biquanimous we have A371731, counted by A371795, even case A006827.
%Y A371955 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371955 A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
%Y A371955 A371783 counts k-quanimous partitions.
%Y A371955 Cf. A000005, A002221, A035470, A064914, A232466, A299701, A321142, A321454, A321455, A326534, A357879, A371781.
%K A371955 nonn
%O A371955 1,1
%A A371955 _Gus Wiseman_, Apr 19 2024