cp's OEIS Frontend

A357976 Numbers with a divisor having the same sum of prime indices as their quotient.

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