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 A350686 #43 Mar 21 2023 06:13:10
%S A350686 12,17,19,20,26,31,211,716,1226,1436,2306,2731,2971,5636,8011,12146,
%T A350686 12721,16921,18266,19441,24481,24691,25796,28316,30026,34651,35876,
%U A350686 37171,45986,49681,51691,56036,58676,61561,67531,77276,98731,98996,104161,104756,108571
%N A350686 Numbers k such that tau(k) + tau(k+1) + tau(k+2) + tau(k+3) = 16, where tau is the number of divisors function A000005.
%C A350686 It can be shown that if tau(k) + tau(k+1) + tau(k+2) + tau(k+3) = 16, the quadruple (tau(k), tau(k+1), tau(k+2), tau(k+3)) must be one of the following, each of which might plausibly occur infinitely often:
%C A350686 (2, 4, 4, 6), which first occurs at k = 12721, 16921, 19441, 24481, ... (A163573);
%C A350686 (2, 6, 4, 4), which first occurs at k = 19, 31, 211, 2731, ...;
%C A350686 (4, 4, 6, 2), which first occurs at k = 26, 1226, 2306, 12146, ...;
%C A350686 (6, 4, 4, 2), which first occurs at k = 20, 716, 1436, 5636, ...; ({A247347(n)-3}, other than its first term)
%C A350686 or one of the following, each of which occurs only once:
%C A350686 (2, 6, 2, 6), which occurs only at k = 17; and
%C A350686 (6, 2, 4, 4), which occurs only at k = 12.
%C A350686 Tau(k) + tau(k+1) + tau(k+2) + tau(k+3) >= 16 for all sufficiently large k; the only numbers k for which tau(k) + tau(k+1) + tau(k+2) + tau(k+3) < 16 are 1..11, 13, 14, and 16.
%H A350686 Jon E. Schoenfield, <a href="/A350686/b350686.txt">Table of n, a(n) for n = 1..10000</a>
%F A350686 { k : tau(k) + tau(k+1) + tau(k+2) + tau(k+3) = 16 }.
%e A350686 The table below includes all terms k such that at least one of the four numbers k, k+1, k+2, k+3 has no prime factor > 5; each such number appears in parentheses in the columns under "factorization".
%e A350686 The table also includes, for each of the patterns (tau(k), tau(k+1), tau(k+2), tau(k+3)) that continues to appear for large k, the smallest such k for which each of the four numbers k, k+1, k+2, k+3 has a prime factor > 5. For each such quadruple, each of the four numbers is the product of a distinct multiplier m from 1..4 and a prime > 5, and each pattern corresponds to a distinct value of k mod 120: the tau patterns (2, 4, 4, 6), (2, 6, 4, 4), (4, 4, 6, 2), and (6, 4, 4, 2) correspond to k mod 120 = 1, 91, 26, and 116, respectively.
%e A350686 .
%e A350686 factorization as
%e A350686 # divisors of m*(prime > 5)
%e A350686 n a(n)=k k k+1 k+2 k+3 k k+1 k+2 k+3 k mod 120
%e A350686 - ------ --- --- --- --- --- --- --- --- ---------
%e A350686 1 12 6 2 4 4 (12) q 2r 3s 12
%e A350686 2 17 2 6 2 6 p (18) r 4s 17
%e A350686 3 19 2 6 4 4 p (20) 3r 2s 19
%e A350686 4 20 6 4 4 2 (20) 3q 2r s 20
%e A350686 5 26 4 4 6 2 2p (27) 4r s 26
%e A350686 6 31 2 6 4 4 p (32) 3r 2s 31
%e A350686 7 211 2 6 4 4 p 4q 3r 2s 91
%e A350686 8 716 6 4 2 2 4p 3q 2r s 116
%e A350686 9 1226 4 4 6 2 2p 3q 4r s 26
%e A350686 17 12721 2 4 4 6 p 2q 3r 4s 1
%t A350686 Position[Plus @@@ Partition[Array[DivisorSigma[0, #] & , 10^5], 4, 1], 16] // Flatten (* _Amiram Eldar_, Jan 12 2022 *)
%o A350686 (PARI) isok(k) = numdiv(k) + numdiv(k+1) + numdiv(k+2) + numdiv(k+3) == 16; \\ _Michel Marcus_, Jan 12 2022
%o A350686 (Python)
%o A350686 from sympy import divisor_count as tau
%o A350686 print([k for k in range( 1, 108572) if tau(k) + tau(k+1) + tau(k+2) + tau(k+3) == 16]) # _Karl-Heinz Hofmann_, Jan 12 2022
%Y A350686 Cf. A000005, A163573, A247347.
%Y A350686 Numbers k such that Sum_{j=0..N-1} tau(k+j) = 2*Sum_{k=1..N} tau(k): A000040 (N=1), A350593 (N=2), A350675 (N=3), (this sequence) (N=4), A350699 (N=5), A350769 (N=6), A350773 (N=7), A350854 (N=8).
%K A350686 nonn
%O A350686 1,1
%A A350686 _Jon E. Schoenfield_, Jan 11 2022