A252424 Numbers k such that sum of odd divisors of k equals sum of squares of primes dividing k.

18, 36, 72, 144, 234, 288, 468, 576, 936, 1152, 1872, 2304, 3744, 4608, 7488, 9216, 14976, 18432, 29952, 36864, 59904, 73728, 119808, 147456, 239616, 294912, 479232, 589824, 958464, 1179648, 1916928, 2359296, 3833856, 4718592, 7667712, 9437184, 15335424, 18874368

Offset

1,1

Comments

Numbers k such that A000593(k) = A005063(k).

a(n) == 0 (mod 18), and the numbers 18*2^m, m = 0,1,... are in the sequence because the odd divisors are {1, 3, 9}, the prime factors are {2, 3} => 2^2 + 3^2 = 1 + 3 + 9 = 13.

The numbers of the form 18*13*2^m are in the sequence because the odd divisors are {1, 3, 9, 13, 39, 117}, the prime factors are {2, 3, 13} => 2^2 + 3^2 + 13^2 = 1 + 3 + 9 + 13 + 39 + 117 = 182.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..56

Example

18 is in the sequence because the prime factors of 18 are {2, 3}, the odd divisors of 18 are {1, 3, 9} => 2^2 + 3^2 = 1 + 3 + 9 = 13.
Or 18 => A000593(18) = A005063(18) = 13.

Maple

with(numtheory):nn:=10^5:
for n from 2 to nn do:
   x:=factorset(n):n0:=nops(x):
   s0:=sum('x[i]^2', 'i'=1..n0):
   y:=divisors(n):n1:=nops(y):
   s :=0 :
        for j from 1 to n1 do :
       if irem (y[j], 2)=1 then s:=s+y[j]:
      else
      fi:
    od:
     if s=s0
    then
   printf(`%d, `, n):
   else
   fi:
od:

Mathematica

a252424[n_Integer] := Module[{f, g},
  f[x_] := Plus @@ Select[Divisors[x], OddQ[#] &];
  g[x_] := Plus @@ (First@Transpose@FactorInteger[x]^2);
Rest@Select[Range[n], f[#] == g[#] &]]; a252424[10^6] (* Michael De Vlieger, Dec 17 2014 *)
Select[Range[19*10^6], Total[Select[Divisors[#], OddQ]]==Total[ FactorInteger[ #][[All, 1]]^2]&] (* Harvey P. Dale, May 11 2020 *)
f[p_, e_] := If[p == 2, 1, (p^(e + 1) - 1)/(p - 1)]; q[n_] := Times @@ f @@@ (fct = FactorInteger[n]) == Total[fct[[;; , 1]]^2]; Select[Range[2, 10^6], q] (* Amiram Eldar, Jul 09 2022 *)

Prog

(PARI) isok(n) = my(f = factor(n)); sum(i=1, #f~, f[i, 1]^2) == sumdiv(n, d, d*(d%2)); \\ Michel Marcus, Dec 17 2014

Crossrefs

Cf. A000593, A005063.

Sequence in context: A083211 A156903 A204824 * A327774 A335784 A347889

Adjacent sequences: A252421 A252422 A252423 * A252425 A252426 A252427

Keyword

nonn

Author

Michel Lagneau, Dec 17 2014

Status

approved