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%I #16 May 31 2022 11:23:04
%S 2,10,14,10,55,91,7,17,19,11,253,13,13,29,31,17,17,703,19,41,43,23,
%T 1081,1,1,53,1,29,1711,1891,31,1,67,1,71,2701,37,1,79,41,3403,43,43,
%U 89,1,47,47,97,1,101,103,53,5671,109,1,113,1,59,59,61,61,1,127,1,131,67,67,137
%N Numerator of sum of all matrix elements M(i,j) = i^2 + j^2 (i,j = 1..n) divided by n!.
%C This is a highly irregular sequence with high points belonging to hexagonal numbers A000384(n) = n*(2n-1) or second hexagonal numbers A014105(n) = n*(2n+1). All other elements of this sequence are equal to 1, n, (n+1) or (2n+1).
%C Numbers n such that a(n) = 1 are {24, 25, 27, 32, 34, 38, 45, 49, 55, 57, 62, 64, 76, 77, 80, 84, 85, 87, 91, 92, 93, 94, ...}. a(n) = n only iff n is prime such that 2n+1 is composite. Such primes (non-Sophie Germain primes) are listed in A053176(n) = {7, 13, 17, 19, 31, 37, 43, 47, 59, 61, 67, 71, 73, 79, 97, ...}. a(n) = n+1 for n = {1, 10, 12, 16, 22, 28, 40, 42, 46, 52, 58, 60, 66, 70, 72, 82, 88, 100, ...}, which coincides with one exception (4) with A109274(n) = {1, 4, 10, 12, 16, 22, 28, 40, 42, 46, 52, 58, 60, ...} Numbers n such that n+1 is prime, 2n+1 composite. a(n) = 2n+1 for n = {8, 9, 14, 15, 20, 21, 26, 33, 35, 39, 44, 48, 50, 51, 54, 56, 63, 65, 68, 69, 74, 75, 81, 86, 90, 95, 98, 99, ...} = A096784(n) Numbers n such that both n and n+1 are composite numbers that sum up to a prime. a(n) = n*(2n+1) for n = {2, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, ...}, which coincides with one exception (3) with A005384(n) = {2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, ...} Sophie Germain primes p: 2p+1 is also prime. a(n) = (n+1)*(2n+1) for n = 6k, where k = {1, 3, 5, 6, 13, 16, 23, 26, 33, 35, 38, 45, 51, 55, 56, 61, 63, 73, 83, 91, 96, 100, ...}. - _Alexander Adamchuk_, Nov 15 2006
%C Numbers n such that a(n) = 1 are listed in A123608(n) = {24, 25, 27, 32, 34, 38, 45, 49, 55, 57, 62, 64, 76, 77, 80, 84, 85, 87, 91, 92, 93, 94, ...} Numbers n such that n, n+1 and 2n+1 are composite. - _Alexander Adamchuk_, Jan 05 2007
%F a(n) = numerator(A098077(n)/n!).
%F a(n) = numerator(n*(n+1)*(2n+1)/3/(n-1)!). - _Alexander Adamchuk_, Nov 15 2006
%e 1/n!*A098077(n) begins 2, 10, 14, 10, 55/12, 91/60, 7/18, 17/210, 19/1344, ... So a(6) = 91.
%t Numerator[Table[1/n!*Sum[Sum[(i^2+j^2), {i, 1, n}], {j, 1, n}], {n, 1, 100}]]
%t Table[ Numerator[ n*(n+1)*(2n+1)/3/(n-1)! ], {n,1,100} ] (* _Alexander Adamchuk_, Nov 15 2006 *)
%o (PARI) a(n) = numerator(sum(i=1, n, sum(j=1, n, i^2 + j^2))/n!); \\ _Michel Marcus_, May 31 2022
%Y Cf. A098077.
%Y Cf. A000384, A014105.
%Y Cf. A053176, A109274, A096784, A005384.
%Y Cf. A123608 (numbers n such that n, n+1 and 2n+1 are composite).
%K nonn
%O 1,1
%A _Alexander Adamchuk_, Oct 28 2004
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