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%I #20 Jul 09 2022 11:08:59
%S 18,36,72,144,234,288,468,576,936,1152,1872,2304,3744,4608,7488,9216,
%T 14976,18432,29952,36864,59904,73728,119808,147456,239616,294912,
%U 479232,589824,958464,1179648,1916928,2359296,3833856,4718592,7667712,9437184,15335424,18874368
%N Numbers k such that sum of odd divisors of k equals sum of squares of primes dividing k.
%C Numbers k such that A000593(k) = A005063(k).
%C 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.
%C 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.
%H Robert G. Wilson v, <a href="/A252424/b252424.txt">Table of n, a(n) for n = 1..56</a>
%e 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.
%e Or 18 => A000593(18) = A005063(18) = 13.
%p with(numtheory):nn:=10^5:
%p for n from 2 to nn do:
%p x:=factorset(n):n0:=nops(x):
%p s0:=sum('x[i]^2','i'=1..n0):
%p y:=divisors(n):n1:=nops(y):
%p s :=0 :
%p for j from 1 to n1 do :
%p if irem (y[j],2)=1 then s:=s+y[j]:
%p else
%p fi:
%p od:
%p if s=s0
%p then
%p printf(`%d, `,n):
%p else
%p fi:
%p od:
%t a252424[n_Integer] := Module[{f, g},
%t f[x_] := Plus @@ Select[Divisors[x], OddQ[#] &];
%t g[x_] := Plus @@ (First@Transpose@FactorInteger[x]^2);
%t Rest@Select[Range[n], f[#] == g[#] &]]; a252424[10^6] (* _Michael De Vlieger_, Dec 17 2014 *)
%t Select[Range[19*10^6],Total[Select[Divisors[#],OddQ]]==Total[ FactorInteger[ #][[All,1]]^2]&] (* _Harvey P. Dale_, May 11 2020 *)
%t 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 *)
%o (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
%Y Cf. A000593, A005063.
%K nonn
%O 1,1
%A _Michel Lagneau_, Dec 17 2014
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