A345473 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).

6, 56, 236, 956, 2636, 3356, 6236, 9716, 10196, 13436, 15896, 18296, 24716, 26396, 36116, 36956, 37196, 42956, 53036, 69356, 82556, 84536, 119516, 121496, 181556, 201116, 204236, 221756, 252116, 259676, 332636, 359036, 365036, 401516

Offset: 1

Author

David Lovler, Jun 21 2021

Keywords

• nonn

Comments

U(n,k) is a commutative and associative array with integer values that depend on whether n and k are odd or even.

U(n,k) = (5*n*k - 3*(n+k-1))/2 when n and k are both odd.

U(n,k) = (5*n*k - 3*n)/2 when n is even and k is odd.

U(n,k) = (5*n*k - 3*k)/2 when n is odd and k is even.

U(n,k) = 5*n*k/2 when n and k are both even.

U(n,1) = n for all n (identity element).

U(n,0) = 0 for all n.

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.

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).

There are 119 terms up to 5*10^6.

Examples

			Array U(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2  10  12  20  22  30  32  40  42  50
   3  12  15  24  27  36  39  48  51  60
   4  20  24  40  44  60  64  80  84 100
   5  22  27  44  49  66  71  88  93 110
   6  30  36  60  66  90  96 120 126 150
   7  32  39  64  71  96 103 128 135 160
   8  40  48  80  88 120 128 160 168 200
   9  42  51  84  93 126 135 168 177 210
  10  50  60 100 110 150 160 200 210 250
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.
		

Links

• David Lovler, Table of n, a(n) for n = 1..119

Crossrefs

In A327263 U(n,k) is called U(5;n,k).

Cf. A340748, A345357, A345474.

Programs

PARI

T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
U(n, k) = (5*n*k - 3*T319929(n, k))/2;
list(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, U(n, k)); ); ); setminus([1..nn], Set(list)); }
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, ", ")); ); }