A019507 Droll numbers: numbers > 1 whose sum of even prime factors equals the sum of odd prime factors.

72, 240, 672, 800, 2240, 4224, 5184, 6272, 9984, 14080, 17280, 33280, 39424, 48384, 52224, 57600, 93184, 116736, 161280, 174080, 192000, 247808, 304128, 373248, 389120, 451584, 487424, 537600, 565248, 585728, 640000, 718848, 1013760, 1089536, 1244160, 1384448

Offset: 1

Author

Mario Velucchi (mathchess(AT)velucchi.it)

Keywords

• nonn

Examples

			6272 = 2*2*2*2*2*2*2*7*7 is droll since 2+2+2+2+2+2+2 = 14 = 7+7.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000
• Giovanni Resta, Droll numbers, Numbers Aplenty.

Crossrefs

For count instead of sum we have A072978.

Partitions of this type are counted by A239261, without zero terms A249914.

For prime indices instead of factors we have A366748, zeros of A366749.

The LHS is A366839 with alternating zeros, for indices A366531, triangle A113686.

The RHS is A366840, for indices A366528, triangle A113685.

A000009 counts partitions into odd parts, ranks A066208.

A035363 counts partitions into even parts, ranks A066207.

A112798 lists prime indices, length A001222, sum A056239.

A257991 counts odd prime indices, even A257992.

A300061 lists numbers with even sum of prime indices, odd A300063.

Cf. A028260, A045931, A100367, A106529, A195017, A239241, A241638, A325698, A325700, A366533.

Programs

Maple

f:= proc(k, m) # numbers whose sum of prime factors >= m is k; m is prime
   local S,p,j;
   option remember;
   if k = 0 then return [1]
   elif m > k then return []
   fi;
   S:= NULL:
   p:= nextprime(m);
   for j from k by -m to 0 do
     S:= S, op(map(`*`,  procname(j,p) , m^((k-j)/m)))
   od;
   [S]
end proc:
g:= proc(N) local m,R;
  R:= NULL;
  for m from 1 while 2^m < N do
   R:= R, op(map(`*`,select(`<=`,f(2*m,3), N/2^m),2^m));
  od;
  sort([R])
end proc:
g(10^8); # Robert Israel, Feb 20 2025

Mathematica

Select[Range[2, 2*10^6, 2], First[#] == Total[Rest[#]] & [Times @@@ FactorInteger[#]] &] (* Paolo Xausa, Feb 19 2025 *)

PARI

isok(n) = {if (n % 2, return (0)); f = factor(n); return (2*f[1,2] == sum(i=2, #f~, f[i,1]*f[i,2]));} \\ Michel Marcus, Jun 21 2013

Formula

These are even numbers k such that A366839(k/2) = A366840(k). - Gus Wiseman, Oct 25 2023 (corrected Feb 19 2025)

Extensions

Name edited by Paolo Xausa, Feb 19 2025