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 A350769 #29 Jan 20 2022 00:45:53
%S A350769 27,28,30,37,38,41,42,57,18362,2914913,5516281,6618242,7224834,
%T A350769 9018353,9339114,10780554,16831081,17800553,18164161,18646202,
%U A350769 20239913,29743561,32464433,32915513,42464514,43502033,45652314,51755761,53464314,62198634,69899754
%N A350769 Numbers k such that tau(k) + ... + tau(k+5) = 28, where tau is the number of divisors function A000005.
%C A350769 It can be shown that if tau(k) + ... + tau(k+5) = 28, the sextuple (tau(k), tau(k+1), tau(k+2), tau(k+3), tau(k+4), tau(k+5)) must be one of the following, each of which might plausibly occur infinitely often:
%C A350769 (2, 4, 4, 6, 4, 8), which first occurs at k = 5516281, 16831081, 18164161, ... (A208455);
%C A350769 (2, 8, 4, 6, 4, 4), which first occurs at k = 2914913, 9018353, 17800553, ...;
%C A350769 (4, 4, 6, 4, 8, 2), which first occurs at k = 18362, 6618242, 18646202, ...;
%C A350769 (8, 4, 6, 4, 4, 2), which first occurs at k = 7224834, 9339114, 10780554, ...;
%C A350769 or one of the following, each of which occurs only once:
%C A350769 (4, 6, 2, 8, 2, 6), which occurs only at k = 27;
%C A350769 (6, 2, 8, 2, 6, 4), which occurs only at k = 28;
%C A350769 (8, 2, 6, 4, 4, 4), which occurs only at k = 30;
%C A350769 (2, 4, 4, 8, 2, 8), which occurs only at k = 37;
%C A350769 (4, 4, 8, 2, 8, 2), which occurs only at k = 38;
%C A350769 (2, 8, 2, 6, 6, 4), which occurs only at k = 41;
%C A350769 (8, 2, 6, 6, 4, 2), which occurs only at k = 42;
%C A350769 (4, 4, 2, 12, 2, 4), which occurs only at k = 57.
%C A350769 The terms of the repeatedly occurring patterns form sequence A071368.
%C A350769 Tau(k) + ... + tau(k+5) >= 28 for all sufficiently large k; the only numbers k for which tau(k) + ... + tau(k+5) < 28 are 1..26, 29, 33, and 34.
%H A350769 Jon E. Schoenfield, <a href="/A350769/b350769.txt">Table of n, a(n) for n = 1..10000</a>
%F A350769 { k : Sum_{j=0..5} tau(k+j) = 28 }.
%e A350769 The table below lists each term k with a pattern (tau(k), ..., tau(k+5)) that appears only once (these appear at n = 1..8) as well as each term k that is the smallest one having a pattern that appears repeatedly for large k. (a(12)=6618242 is omitted from the table because it has the same pattern as a(9)=18362.)
%e A350769 Each of the repeatedly occurring patterns corresponds to one of the four possible orders in which the multipliers m=1..6 can appear among 6 consecutive integers of the form m*prime, and thus to a single residue of k modulo 2520; e.g., k=18362 begins a run of 6 consecutive integers having the form (2*p, 3*q, 4*r, 5*s, 6*t, 1*u), where p, q, r, s, t, and u are distinct primes > 6, and all such runs satisfy k == 722 (mod 2520).
%e A350769 For each of the patterns that does not occur repeatedly, one or more of the six consecutive integers in k..k+5 has no prime factor > 6; each such integer appears in parentheses in the "factorization" columns.
%e A350769 .
%e A350769 . factorization as k
%e A350769 # divisors of m*(prime > 6) mod
%e A350769 n a(n)=k k k+1 k+2 k+3 k+4 k+5 k k+1 k+2 k+3 k+4 k+5 2520
%e A350769 - -------- --- --- --- --- --- --- --- --- --- --- --- --- ----
%e A350769 1 27 4 6 2 8 2 6 (27) 4q r (30) t (32) 27
%e A350769 2 28 6 2 8 2 6 4 4p q (30) s (32) 3u 28
%e A350769 3 30 8 2 6 4 4 4 (30) q (32) 3s 2t 5u 30
%e A350769 4 37 2 4 4 8 2 8 p 2q 3r (40) t 6u 37
%e A350769 5 38 4 4 8 2 8 2 2p 3q (40) s 6t u 38
%e A350769 6 41 2 8 2 6 6 4 p 6q r 4s (45) 2u 41
%e A350769 7 42 8 2 6 6 4 2 6p q 4r (45) 2t u 42
%e A350769 8 57 4 4 2 12 2 4 3p 2q r (60) t 2u 57
%e A350769 9 18362 4 4 6 4 8 2 2p 3q 4r 5s 6t u 722
%e A350769 10 2914913 2 8 4 6 4 4 p 6q 5r 4s 3t 2u 1793
%e A350769 11 5516281 2 4 4 6 4 8 p 2q 3r 4s 5t 6u 1
%e A350769 13 7224834 8 4 6 4 4 2 6p 5q 4r 3s 2t u 2514
%t A350769 Position[Plus @@@ Partition[Array[DivisorSigma[0, #] &, 10^7], 6, 1], 28] // Flatten (* _Amiram Eldar_, Jan 16 2022 *)
%o A350769 (Python) from sympy import divisor_count as tau
%o A350769 taulist = [1, 2, 2, 3, 2, 4]
%o A350769 for k in range(1, 10000000):
%o A350769 if sum(taulist) == 28: print(k, end=", ")
%o A350769 taulist.append(tau(k+6))
%o A350769 del taulist[0] # _Karl-Heinz Hofmann_, Jan 18 2022
%Y A350769 Cf. A000005, A071368, A208455.
%Y A350769 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), A350686 (N=4), A350699 (N=5), (this sequence) (N=6), A350773 (N=7), A350854 (N=8).
%K A350769 nonn
%O A350769 1,1
%A A350769 _Jon E. Schoenfield_, Jan 14 2022