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A371010
Powerful numbers that are the sum of 2 squares.
2
1, 4, 8, 9, 16, 25, 32, 36, 49, 64, 72, 81, 100, 121, 125, 128, 144, 169, 196, 200, 225, 256, 288, 289, 324, 361, 392, 400, 441, 484, 500, 512, 529, 576, 625, 648, 676, 729, 784, 800, 841, 900, 961, 968, 1000, 1024, 1089, 1125, 1152, 1156, 1225, 1296, 1352, 1369
OFFSET
1,2
COMMENTS
Each term can be decomposed in a unique way as 2^m * i * j^2 where m >= 2, i is a powerful number whose prime factors are all of the form 4*k + 1 (A369563), and j is a number whose prime factors are all of the form 4*k + 3 (A004614).
FORMULA
The number of terms that do not exceed x is ~ c * sqrt(x), where c = (6/Pi^2) * (1 + 1/(3*(sqrt(2)-1))) * Product_{primes p == 1 (mod 4)} (1 + 1/((sqrt(p)-1)*(p+1))) * Product_{primes p == 3 (mod 4)} (1 + 1/(p^2-1)) = 1.58769... (Jakimczuk, 2024, Theorem 4.7, p. 50).
Sum_{n>=1} 1/a(n) = (3/2) * Product_{primes p == 1 (mod 4)} (1 + 1/(p*(p-1))) * Product_{primes p == 3 (mod 4)} (1 + 1/(p^2-1)) = (3*Pi^2/16) * A334424 = 1.86676402705119927669... .
MATHEMATICA
Select[Range[1500], SquaresR[2, #] > 0 && (# == 1 || Min[FactorInteger[#][[;; , 2]]] > 1) &]
PROG
(PARI) is(n) = {my(f=factor(n)); for(i=1, #f~, if(f[i, 2] == 1 || (f[i, 2]%2 && f[i, 1]%4 == 3), return(0))); 1; }
CROSSREFS
Intersection of A001481 and A001694.
A371011 is a subsequence.
Sequence in context: A010417 A155568 A353485 * A067252 A348995 A324723
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 08 2024
STATUS
approved