A252044 Numbers n such that s + 1/p = 0, where {d(i), i=1..q} are the q distinct prime divisors of n, s = Sum_{i=1..q} (-1)^(i+1)*i/d(i) and p = Product_{i=1..q} d(i).

6, 12, 15, 18, 24, 36, 45, 48, 54, 72, 75, 91, 96, 108, 114, 135, 144, 162, 192, 216, 225, 228, 288, 324, 342, 375, 384, 405, 432, 456, 486, 576, 637, 648, 675, 684, 703, 768, 864, 912, 972, 1026, 1125, 1152, 1183, 1215, 1296, 1368, 1458, 1536, 1728, 1824, 1875

Offset: 1

Author

Michel Lagneau, Dec 13 2014

Keywords

• nonn

Comments

The semiprimes p*q, p and q prime with q=2*p-1 (A129521) are in the sequence.

Examples

			18 is in the sequence because the prime factors of 18 are {2,3} => s = 1/2 - 2/3, 1/p = 1/6 and 1/2 - 2/3 + 1/6 = -1/6 + 1/6 = 0.
114 is in the sequence because the prime factors of 114 are {2,3,19} => s = 1/2 - 2/3 + 3/19, 1/p = 1/114 and 1/2 - 2/3 + 3/19 + 1/114 = -1/114 + 1/114 = 0.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A129521, A007947 (product of the distinct prime factors of n).

Programs

Maple

with(numtheory):nn:=10000:
for n from 1 to nn do:
   x:=factorset(n):n0:=nops(x):
   s:=sum('i*((-1)^(i+1))/x[i]','i'=1..n0):s0:=product('x[i]','i'=1..n0):
   p:=product('x[i]','i'=1..n0):s2:=s+1/s0:
    if s2=0
    then
    printf(`%d, `,n):
    else
    fi:
  od:

Mathematica

fQ[n_] := Block[{pd = First@# & /@ FactorInteger@ n, rng}, rng = Range@ Length@ pd; 1 == (Times @@ pd)*Total[rng/pd*((-1)^rng)]]; Select[ Range@ 2000, fQ@# &] (* Robert G. Wilson v, Jan 11 2015 *)

PARI

isok(n) = {my(vp = factor(n)[,1]~); 1/prod(i=1, #vp, vp[i]) + sum(i=1, #vp, (-1)^(i+1)*i/vp[i]) == 0;} \\ Michel Marcus, Jan 12 2015