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%I #12 Oct 26 2023 20:16:18
%S 1,4,9,12,16,25,30,36,40,48,49,63,64,70,81,84,90,100,108,112,120,121,
%T 144,154,160,165,169,192,196,198,210,220,225,252,256,264,270,273,280,
%U 286,289,300,324,325,336,351,352,360,361,364,390,400,432,441,442,448
%N Numbers with a divisor having the same sum of prime indices as their quotient.
%C 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 Robert Israel, <a href="/A357976/b357976.txt">Table of n, a(n) for n = 1..10000</a>
%e The terms together with their prime indices begin:
%e 1: {}
%e 4: {1,1}
%e 9: {2,2}
%e 12: {1,1,2}
%e 16: {1,1,1,1}
%e 25: {3,3}
%e 30: {1,2,3}
%e 36: {1,1,2,2}
%e 40: {1,1,1,3}
%e 48: {1,1,1,1,2}
%e 49: {4,4}
%e For example, 40 has factorization 8*5, and both factors have the same sum of prime indices 3, so 40 is in the sequence.
%p filter:= proc(n) local F,s,t,i,R;
%p F:= ifactors(n)[2];
%p F:= map(t -> [numtheory:-pi(t[1]),t[2]], F);
%p s:= add(t[1]*t[2],t=F)/2;
%p if not s::integer then return false fi;
%p try
%p R:= Optimization:-Maximize(0, [add(F[i][1]*x[i],i=1..nops(F)) = s, seq(x[i]<= F[i][2],i=1..nops(F))], assume=nonnegint, depthlimit=20);
%p catch "no feasible integer point found; use feasibilitytolerance option to adjust tolerance": return false;
%p end try;
%p true
%p end proc:
%p filter(1):= true:
%p select(filter, [$1..1000]); # _Robert Israel_, Oct 26 2023
%t sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
%t Select[Range[100],MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]
%Y The partitions with these Heinz numbers are counted by A002219.
%Y A subset of A300061.
%Y The squarefree case is A357854, counted by A237258.
%Y Positions of nonzero terms in A357879.
%Y A001222 counts prime factors, distinct A001221.
%Y A056239 adds up prime indices, row sums of A112798.
%Y Cf. A033879, A033880, A064914, A181819, A213086, A235130, A237194, A276107, A300273, A321144, A357975.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 26 2022
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