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A252424
Numbers k such that sum of odd divisors of k equals sum of squares of primes dividing k.
1
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
Sequence in context: A083211 A156903 A204824 * A327774 A335784 A347889
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 17 2014
STATUS
approved