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 A347719 #18 Oct 05 2021 03:33:42 %S A347719 4,165,2275,18473,45617,71383,257393,257393,1239907,1275797,1851847, %T A347719 4411843,6865337,6865337,8312467,15763207,24157963,33684317,33684317, %U A347719 60428597,61182103,61694813,73803517,104622971,128397967,128397967,173805187,214820797,284708981 %N A347719 a(n) is the smallest prime(n)-rough number k that has more divisors than k-1 and more divisors than k+1. %C A347719 Equivalently, a(n) is the smallest number k that is not divisible by any of the first n-1 primes such that d(k-1) < d(k) > d(k+1), where d(k) = A000005(k) is the number of divisors of k. %C A347719 a(1) = 4 is the initial term of A075027: 4 is the smallest k such that d(k-1) < d(k) > d(k+1). %C A347719 a(2) = 165 = A323379(1) is the smallest odd k such that d(k-1) < d(k) > d(k+1). %C A347719 For n > 2, since neither 2 nor 3 divides k, one of k's two nearest neighbors, k-1 and k+1, is a multiple of 3, and both of those neighbors are even numbers, so one of those neighbors is a multiple of 6, and thus (since that neighbor is not 6, 12, or 18) that neighbor has at least 8 divisors, so k must have at least 9 divisors, so k must be the product of at least 4 primes (counted with multiplicity). For distinct primes p, q, r, and s, the product k cannot be p^4 (only 5 divisors) or p^3 * q (only 8 divisors), nor can it be p^2 * q^2 (which would have 9 divisors, but since k would be an odd square not divisible by 3, k-1 would be a proper multiple of 24 and would thus have more than 9 divisors), so k must be p^2 * q * r (12 divisors) or p*q*r*s (16 divisors). %C A347719 For n > 3, k cannot simultaneously be congruent to +-1 (mod 10) and be of the form p^2 * q * r: one of its two neighbors would be divisible by 4 and the other twice an odd number, one would be divisible by 3, and one would be divisible by 5, so at least one of the two would have at least 12 divisors. %C A347719 It appears that the least prime factor of a(n) is usually the n-th prime, but there are 299 exceptions among the first 1000 terms, beginning with the terms for n = 7, 13, and 18 (see the Example section). %C A347719 Conjecture: no term is the product of more than 4 prime factors, counted with multiplicity. %e A347719 In the table below, an asterisk after the number in the "n-th prime" column appears in the row for each number n such that the least prime factor of a(n) is not the n-th prime. %e A347719 . %e A347719 number of divisors of %e A347719 ====================== n-th prime factorization %e A347719 n a(n) a(n)-1 a(n) a(n)+1 prime of a(n) %e A347719 -- -------- ------ ------ ------ ----- ------------------- %e A347719 1 4 2 3 2 2 2 * 2 %e A347719 2 165 6 8 4 3 3 * 5 * 11 %e A347719 3 2275 8 12 6 5 5 * 5 * 7 * 13 %e A347719 4 18473 8 12 8 7 7 * 7 * 13 * 29 %e A347719 5 45617 10 12 8 11 11 * 11 * 13 * 29 %e A347719 6 71383 8 12 8 13 13 * 17 * 17 * 19 %e A347719 7 257393 10 12 8 17* 19 * 19 * 23 * 31 %e A347719 8 257393 10 12 8 19 19 * 19 * 23 * 31 %e A347719 9 1239907 8 16 6 23 23 * 31 * 37 * 47 %e A347719 10 1275797 6 12 8 29 29 * 29 * 37 * 41 %e A347719 11 1851847 8 12 8 31 31 * 31 * 41 * 47 %e A347719 12 4411843 8 16 12 37 37 * 43 * 47 * 59 %e A347719 13 6865337 8 12 8 41* 43 * 43 * 47 * 79 %e A347719 14 6865337 8 12 8 43 43 * 43 * 47 * 79 %e A347719 15 8312467 8 12 6 47 47 * 47 * 53 * 71 %e A347719 16 15763207 8 12 8 53 53 * 59 * 71 * 71 %e A347719 17 24157963 12 16 6 59 59 * 71 * 73 * 79 %e A347719 18 33684317 6 16 12 61* 67 * 71 * 73 * 97 %Y A347719 Cf. A000005, A075027, A323379. %K A347719 nonn %O A347719 1,1 %A A347719 _Jon E. Schoenfield_, Sep 26 2021