A345474 Given the associative array U(n,k) described below, numbers m > 7 such that [m-5..m+5] are not in U(n,k) (excluding the first row and column).

8, 106, 26874, 105834, 1080234

Offset: 1

Author

David Lovler, Jun 22 2021

Keywords

• nonn
• more

Comments

There are no more terms up to 5*10^6.

All terms equal 1 (mod 7).

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) = (7*n*k - 5*(n+k-1))/2 when n and k are both odd,

U(n,k) = (7*n*k - 5*n)/2 when n is even and k is odd,

U(n,k) = (7*n*k - 5*k)/2 when n is odd and k is even and

U(n,k) = 7*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.

U(n,k) can be expressed as (7*n*k - 5*U(0;n,k))/2, where U(0;n,k) has four cases.

U(0;n,k) = n+k-1 when n and k are both odd,

U(0;n,k) = n when n is even and k is odd,

U(0;n,k) = k when n is odd and k is even and

U(0;n,k) = 0 when n and k are both even.

The ordered list of numbers > 7 that do not appear in array U(n,k) for n and k > 1 can have at most 5 consecutive even numbers and at most 7 consecutive odd numbers. See rows 2 and 3.

U(n,k) is part of a hierarchy of multiplication-like arrays, mentioned in A327263, in which the entries depend on the parity of n and k. U(0;n,k), which is A319929(n,k), is their common parity-dependent component. For i >= 0, U(i;n,k) = (i*n*k - (i-2)*U(0;n,k))/2. In the current sequence U(n,k) = U(7;n,k). Each of these arrays leaves behind a list of numbers that do not appear outside of row 1 and column 1. Think of the prime number sieve.

U(2;n,k) is normal multiplication. For i > 2, these lists are progressively more dense and include even numbers as well as odd numbers. This allows strings of consecutive integers. Here we are interested in the maximum length strings for each i.

The ordered list of numbers > i that do not appear in array U(i;n,k) for n and k > 1 can have at most i-2 consecutive even numbers and at most i consecutive odd numbers. To verify this, look at rows 2 and 3. i consecutive odd numbers cannot interleaf with i-2 even numbers, but i-1 odd numbers can. Thus the longest strings for each i are of length 2i-3. When i is even, m is odd. When i is odd, m is even.

In general, there are more terms when i is odd than when i is even. This is because there are a few ways that i consecutive odd number can overlap i-2 consecutive even numbers.

For this sequence and similar sequences constructed from U(i;n,k), all terms m == 1 (mod i). To prove this, look at gaps of 2i-3 in row 2 of U(i;n,k). The longest strings of consecutive numbers not in U(i;n,k) can occur only for these 2i-3 numbers. The second number before any of the gaps is an even number of the form U(i;2,e) = i*2*e/2 == 0 (mod i) (where e is an even number). The middle of the string, m = U(i;2,e) + i + 1. Thus m == 1 (mod i).

Likewise, for any i, all terms m == 2 (mod i+1).

The following observations are included to expand upon the theme. For odd i in [9..23], the number of terms in [i+2..5*10^6] represented as [i, number of terms] are [9, 3], [11, 8], [13, 3], [15, 0], [17, 3], [19, 3], [21, 0], [23, 3]. For odd i in [25..99], 11 i's have no terms in [i+2..5*10^5], 12 have 1 such term, 9 have 2, 4 have 3, 1 has 4 and 1 has 5 such terms.

The scarcity of terms for even i's is borne out by the observation that up to 5*10^6, for even i in [6..22], the only terms > i+1 occur when i = 14 (1667), i = 20 (3341 and 1663181) and i = 22 (16171). Continuing the observation for even i in [24..140], up to 5*10^5 only 14 i's have a term > i+1.

Examples

			Array U(0;n,k) = A319929(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   0   2   0   2   0   2   0   2   0
   3   2   5   4   7   6   9   8  11  10
   4   0   4   0   4   0   4   0   4   0
   5   2   7   4   9   6  11   8  13  10
   6   0   6   0   6   0   6   0   6   0
   7   2   9   4  11   6  13   8  15  10
   8   0   8   0   8   0   8   0   8   0
   9   2  11   4  13   6  15   8  17  10
  10   0  10   0  10   0  10   0  10   0
Array U(n,k) = U(7;n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2  14  16  28  30  42  44  56  58  70
   3  16  19  32  35  48  51  64  67  80
   4  28  32  56  60  84  88 112 116 140
   5  30  35  60  65  90  95 120 125 150
   6  42  48  84  90 126 132 168 174 210
   7  44  51  88  95 132 139 176 183 220
   8  56  64 112 120 168 176 224 232 280
   9  58  67 116 125 174 183 232 241 290
  10  70  80 140 150 210 220 280 290 350
Numbers up to 200 not in U(n,k) (excluding row 1 and column 1): 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 33, 34, 36, 37, 38, 39, 40, 41, 43, 45, 46, 47, 49, 50, 52, 53, 54, 55, 57, 59, 61, 62, 63, 66, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 85, 87, 89, 91, 92, 93, 94, 97, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 117, 118, 119, 121, 122, 123, 124, 127, 129, 130, 133, 134, 135, 136, 137, 138, 141, 143, 145, 146, 148, 149, 151, 152, 153, 157, 158, 159, 161, 162, 164, 165, 166, 167, 169, 171, 173, 175, 177, 178, 181, 186, 187, 188, 189, 190, 191, 193, 194, 197, 199.
		

Links

• David Lovler, Terms for i <= 140

Crossrefs

In A327263 U(n,k) is called U(7;n,k).

Cf. A340748, A345357, A345473.

Programs

PARI

T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
U(n, k) = (7*n*k - 5*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=8, #v-1, my(x=v[m]); if (#setintersect(v, [x-5..x+5])==11, print1(x, ", ")); ); }

PARI

/* This program computes terms of sequences based on U(i;n,k) for i >= 2. */
T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
U(n, k) = (i*n*k - (i-2)*T319929(n, k))/2; \\ U(i; n, k)
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=i+1, #v-1, my(x=if(Mod(v[m],i)==1,v[m])); if (#setintersect(v,[x-i+2..x+i-2])==2*i-3, print1(x, ", ")); ); }
/* Type for example: i=4; lista(10^6) */