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A252424 Numbers n such that sum of odd divisors of n equals sum of squares of primes dividing n. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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 *)

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 * A115550 A061713 A198802

Adjacent sequences:  A252421 A252422 A252423 * A252425 A252426 A252427

KEYWORD

nonn

AUTHOR

Michel Lagneau, Dec 17 2014

STATUS

approved

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Last modified May 26 17:40 EDT 2019. Contains 323597 sequences. (Running on oeis4.)