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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 4 19:24 EDT 2023. Contains 363128 sequences. (Running on oeis4.)