This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A345474 #38 Jan 25 2022 08:48:36
%S A345474 8,106,26874,105834,1080234
%N 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).
%C A345474 There are no more terms up to 5*10^6.
%C A345474 All terms equal 1 (mod 7).
%C A345474 U(n,k) is a commutative and associative array with integer values that depend on whether n and k are odd or even.
%C A345474 U(n,k) = (7*n*k - 5*(n+k-1))/2 when n and k are both odd,
%C A345474 U(n,k) = (7*n*k - 5*n)/2 when n is even and k is odd,
%C A345474 U(n,k) = (7*n*k - 5*k)/2 when n is odd and k is even and
%C A345474 U(n,k) = 7*n*k/2 when n and k are both even.
%C A345474 U(n,1) = n for all n (identity element).
%C A345474 U(n,0) = 0 for all n.
%C A345474 U(n,k) can be expressed as (7*n*k - 5*U(0;n,k))/2, where U(0;n,k) has four cases.
%C A345474 U(0;n,k) = n+k-1 when n and k are both odd,
%C A345474 U(0;n,k) = n when n is even and k is odd,
%C A345474 U(0;n,k) = k when n is odd and k is even and
%C A345474 U(0;n,k) = 0 when n and k are both even.
%C A345474 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.
%C A345474 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.
%C A345474 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.
%C A345474 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.
%C A345474 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.
%C A345474 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).
%C A345474 Likewise, for any i, all terms m == 2 (mod i+1).
%C A345474 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.
%C A345474 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.
%H A345474 David Lovler, <a href="/A345474/a345474.txt">Terms for i <= 140</a>
%e A345474 Array U(0;n,k) = A319929(n,k) begins:
%e A345474 1 2 3 4 5 6 7 8 9 10
%e A345474 2 0 2 0 2 0 2 0 2 0
%e A345474 3 2 5 4 7 6 9 8 11 10
%e A345474 4 0 4 0 4 0 4 0 4 0
%e A345474 5 2 7 4 9 6 11 8 13 10
%e A345474 6 0 6 0 6 0 6 0 6 0
%e A345474 7 2 9 4 11 6 13 8 15 10
%e A345474 8 0 8 0 8 0 8 0 8 0
%e A345474 9 2 11 4 13 6 15 8 17 10
%e A345474 10 0 10 0 10 0 10 0 10 0
%e A345474 Array U(n,k) = U(7;n,k) begins:
%e A345474 1 2 3 4 5 6 7 8 9 10
%e A345474 2 14 16 28 30 42 44 56 58 70
%e A345474 3 16 19 32 35 48 51 64 67 80
%e A345474 4 28 32 56 60 84 88 112 116 140
%e A345474 5 30 35 60 65 90 95 120 125 150
%e A345474 6 42 48 84 90 126 132 168 174 210
%e A345474 7 44 51 88 95 132 139 176 183 220
%e A345474 8 56 64 112 120 168 176 224 232 280
%e A345474 9 58 67 116 125 174 183 232 241 290
%e A345474 10 70 80 140 150 210 220 280 290 350
%e A345474 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.
%o A345474 (PARI) T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
%o A345474 U(n, k) = (7*n*k - 5*T319929(n, k))/2;
%o A345474 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 A345474 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, ", ")); ); }
%o A345474 (PARI)
%o A345474 /* This program computes terms of sequences based on U(i;n,k) for i >= 2. */
%o A345474 T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
%o A345474 U(n, k) = (i*n*k - (i-2)*T319929(n, k))/2; \\ U(i; n, k)
%o A345474 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 A345474 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, ", ")); ); }
%o A345474 /* Type for example: i=4; lista(10^6) */
%Y A345474 In A327263 U(n,k) is called U(7;n,k).
%Y A345474 Cf. A340748, A345357, A345473.
%K A345474 nonn,more
%O A345474 1,1
%A A345474 _David Lovler_, Jun 22 2021