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 A350675 #28 Jan 20 2022 00:32:31
%S A350675 6,11,13,17,21,37,57,157,177,381,501,541,717,877,1201,1317,1381,1437,
%T A350675 1621,1821,2017,2557,2577,2857,2901,3061,3117,3777,4281,4357,4441,
%U A350675 4677,4701,5077,5097,5581,5637,5701,5937,6337,6637,6661,6717,6997,7417,8221,8781
%N A350675 Numbers k such that tau(k) + tau(k+1) + tau(k+2) = 10, where tau is the number of divisors function A000005.
%C A350675 Since tau(k) + tau(k+1) + tau(k+2) = 10 and no three consecutive integers include more than one square, the triple (tau(k), tau(k+1), tau(k+2)) must consist of three even numbers, so it must be one of (2, 2, 6), (2, 4, 4), (2, 6, 2), (4, 2, 4), (4, 4, 2), and (6, 2, 2). Of these, (2, 2, 6) and (6, 2, 2) are impossible. Of the remaining patterns:
%C A350675 (2, 4, 4) requires that k be an odd prime other than 3, followed by two semiprimes, so k is a prime p such that (p+1)/2 and (p+2)/3 are also prime, and such primes are 13, 37, 157, 541, ... (A036570);
%C A350675 (2, 6, 2) requires that (k, k+2) be a twin prime pair whose average has exactly 6 divisors, and is thus either 12 or 18, so k is 11 or 17;
%C A350675 (4, 2, 4) requires that k+1 be an odd prime, with both k and k+2 having exactly 4 divisors, even though one of them is a multiple of 4, so that one is k+2 = 2^3 = 8, so k = 6;
%C A350675 (4, 4, 2) requires that k+2 be an odd prime > 3, preceded by two semiprimes, so k+2 is a prime p such that (p-1)/2 and (p-2)/3 are also prime, so k+2 is in {23, 59, 179, 383, ...} (which is A181841, after its first two terms, 7 and 11), so k is in {A181841(n) - 2} \ {5, 9}, i.e., k is in {21, 57, 177, 381, ...}.
%C A350675 Tau(k) + tau(k+1) + tau(k+2) >= 10 for all sufficiently large k; the only numbers k for which tau(k) + tau(k+1) + tau(k+2) < 10 are 1..5, 7, and 9.
%H A350675 Jon E. Schoenfield, <a href="/A350675/b350675.txt">Table of n, a(n) for n = 1..10000</a>
%F A350675 { k : tau(k) + tau(k+1) + tau(k+2) = 10 }.
%F A350675 UNION({6}, {11, 17}, A036570, {A181841(n) - 2} \ {5, 9}).
%F A350675 a(n) = A317670(n) - 1.
%e A350675 Each of the patterns (tau(k), ..., tau(k+2)) that appears repeatedly for large k corresponds to one of the two possible orders in which the multipliers m=1..3 can appear among 3 consecutive integers of the form m*prime. E.g., k=37 begins a run of 3 consecutive integers having the form (p, 2*q, 3*r), where p, q, and r are distinct primes > 3; k=57 begins a similar run, but there the 3 consecutive integers have the form (3*p, 2*q, r).
%e A350675 For each of the patterns of tau values that does not occur repeatedly for large k, one or more of the three consecutive integers in k..k+2 has no prime factor > 3; in the table below, each such integer appears in parentheses in the columns on the right.
%e A350675 .
%e A350675 factorization as
%e A350675 # divisors of m*(prime > 3)
%e A350675 n a(n)=k k k+1 k+2 k k+1 k+2
%e A350675 - ------ --- --- --- ---- ---- ----
%e A350675 1 6 4 2 4 (6) q (8)
%e A350675 2 11 2 6 2 p (12) r
%e A350675 3 13 2 4 4 p 2q 3r
%e A350675 4 17 2 6 2 p (18) r
%e A350675 5 21 4 4 2 3p 2q r
%e A350675 6 37 2 4 4 p 2q 3r
%e A350675 7 57 4 4 2 3p 2q r
%t A350675 Position[Plus @@@ Partition[Array[DivisorSigma[0, #] & , 10^4], 3, 1], 10] // Flatten (* _Amiram Eldar_, Jan 11 2022 *)
%o A350675 (PARI) isok(k) = numdiv(k) + numdiv(k+1) + numdiv(k+2) == 10; \\ _Michel Marcus_, Jan 16 2022
%Y A350675 Cf. A000005, A036570, A181841, A307120, A317670.
%Y A350675 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), (this sequence) (N=3), A350686 (N=4), A350699 (N=5), A350769 (N=6), A350773 (N=7), A350854 (N=8).
%K A350675 nonn
%O A350675 1,1
%A A350675 _Jon E. Schoenfield_, Jan 10 2022