A357976 - OEIS

%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