%I #23 Oct 02 2021 19:15:15
%S 6,56,236,956,2636,3356,6236,9716,10196,13436,15896,18296,24716,26396,
%T 36116,36956,37196,42956,53036,69356,82556,84536,119516,121496,181556,
%U 201116,204236,221756,252116,259676,332636,359036,365036,401516
%N Given the associative array U(n,k) described below, numbers m > 5 such that [m-3..m+3] are not in U(n,k) (excluding the first row and column).
%C U(n,k) is a commutative and associative array with integer values that depend on whether n and k are odd or even.
%C U(n,k) = (5*n*k - 3*(n+k-1))/2 when n and k are both odd.
%C U(n,k) = (5*n*k - 3*n)/2 when n is even and k is odd.
%C U(n,k) = (5*n*k - 3*k)/2 when n is odd and k is even.
%C U(n,k) = 5*n*k/2 when n and k are both even.
%C U(n,1) = n for all n (identity element).
%C U(n,0) = 0 for all n.
%C The ordered list of numbers >5 that do not appear in array U(n,k) for n and k > 1 can have at most 3 consecutive even numbers and at most 5 consecutive odd numbers. See rows 2 and 3.
%C The terms all end in 6 because row 2 of U(n,k) has all numbers that end in 0 or 2 and there are at most 3 consecutive even numbers in the set of numbers not in array U(n,k) excluding the first row and column (see comment for A327263).
%C There are 119 terms up to 5*10^6.
%H David Lovler, <a href="/A345473/b345473.txt">Table of n, a(n) for n = 1..119</a>
%e Array U(n,k) begins:
%e 1 2 3 4 5 6 7 8 9 10
%e 2 10 12 20 22 30 32 40 42 50
%e 3 12 15 24 27 36 39 48 51 60
%e 4 20 24 40 44 60 64 80 84 100
%e 5 22 27 44 49 66 71 88 93 110
%e 6 30 36 60 66 90 96 120 126 150
%e 7 32 39 64 71 96 103 128 135 160
%e 8 40 48 80 88 120 128 160 168 200
%e 9 42 51 84 93 126 135 168 177 210
%e 10 50 60 100 110 150 160 200 210 250
%e Numbers up to 100 not in U(n,k) (excluding row 1 and column 1): 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 16, 17, 18, 19, 21, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 41, 43, 45, 46, 47, 53, 54, 55, 56, 57, 58, 59, 61, 65, 67, 68, 69, 73, 74, 76, 77, 78, 79, 81, 83, 85, 86, 89, 91, 94, 95, 97, 98.
%o (PARI) T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
%o U(n, k) = (5*n*k - 3*T319929(n, k))/2;
%o list(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, U(n, k)); ); ); setminus([1..nn], Set(list)); }
%o lista(nn) = {my(v=Vec(list(nn))); for (m=6, #v-1, my(x=v[m]); if (#setintersect(v,[x-3..x+3])==7, print1(x, ", ")); ); }
%Y In A327263 U(n,k) is called U(5;n,k).
%Y Cf. A340748, A345357, A345474.
%K nonn
%O 1,1
%A _David Lovler_, Jun 21 2021