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%I #32 Mar 13 2023 17:48:46
%S 4,6,16,18,20,32,52,72,102,180,3212
%N Consider all ways of writing 2n = p + q where p, q are primes, p <= n and q >= n; let s1(n) = sum of the p's and s2(n) = sum of the q's; the sequence lists the integers 2n for which s1(n) divides s2(n).
%C This is a list of values of 2n such that A185297(n) divides A187129(n). - _N. J. A. Sloane_, Mar 10 2011
%C I have some fast code for counting Goldbach partitions. I made a slight change so that it sums the partitions instead. Using this new program, I did not find any additional terms < 10^7. - _T. D. Noe_, Mar 10 2011
%e For 2n=52, the partitions are (5,47), (11,41) and (23,29). The lesser sum of primes is 5+11+23=39 and the greater sum of primes is 29+41+47=117, with 39|117 for quotient 3.
%e For the 2n listed, the values of (s1(n), s2(n)/s1(n)) are (2,1), (3,1), (8,3), (12,2), (10,3), (16,3), (39,3), (108,3), (204,3), (630,3), (35332,3).
%t okQ[n_] := Module[{p, q}, p = Select[Prime[Range[PrimePi[n]]], PrimeQ[2 n - #] &]; q = 2 n - p; Mod[Plus @@ q, Plus @@ p] == 0]; 2*Select[Range[2, 10000], okQ]
%o (PARI) isok(n) = if (!(n%2), my(s1=0, s2=0); forprime(p=1, n/2, if (isprime(n-p), s1 += p; s2 += n-p)); s1 && !(s2 % s1));
%o for (n=1, 10000, if (isok(2*n), print1(2*n, ", "))) \\ _Michel Marcus_, Mar 13 2023
%Y Cf. A045917, A185297, A187129.
%K nonn,more
%O 1,1
%A _J. M. Bergot_, Feb 14 2011
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