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

1,1

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.

Links

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

Example

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.

Prog

(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, ", ")); ); }

Crossrefs

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

Cf. A340748, A345357, A345474.

Sequence in context: A164579 A137034 A177059 * A053127 A068495 A166766

Adjacent sequences: A345470 A345471 A345472 * A345474 A345475 A345476

Keyword

nonn

Author

David Lovler, Jun 21 2021

Status

approved