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

1, 4, 9, 12, 16, 25, 30, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 144, 154, 160, 165, 169, 192, 196, 198, 210, 220, 225, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 351, 352, 360, 361, 364, 390, 400, 432, 441, 442, 448

Offset: 1

Author

Gus Wiseman, Oct 26 2022

Keywords

• nonn

Comments

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:
   1: {}
   4: {1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  25: {3,3}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
For example, 40 has factorization 8*5, and both factors have the same sum of prime indices 3, so 40 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

The partitions with these Heinz numbers are counted by A002219.

A subset of A300061.

The squarefree case is A357854, counted by A237258.

Positions of nonzero terms in A357879.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

Cf. A033879, A033880, A064914, A181819, A213086, A235130, A237194, A276107, A300273, A321144, A357975.

Programs

Maple

filter:= proc(n) local F,s,t,i,R;
  F:= ifactors(n)[2];
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], F);
  s:= add(t[1]*t[2],t=F)/2;
  if not s::integer then return false fi;
  try
  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);
  catch "no feasible integer point found; use feasibilitytolerance option to adjust tolerance": return false;
  end try;
  true
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Oct 26 2023

Mathematica

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Select[Range[100],MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]