A347719 a(n) is the smallest prime(n)-rough number k that has more divisors than k-1 and more divisors than k+1.

4, 165, 2275, 18473, 45617, 71383, 257393, 257393, 1239907, 1275797, 1851847, 4411843, 6865337, 6865337, 8312467, 15763207, 24157963, 33684317, 33684317, 60428597, 61182103, 61694813, 73803517, 104622971, 128397967, 128397967, 173805187, 214820797, 284708981

Offset: 1

Author

Jon E. Schoenfield, Sep 26 2021

Keywords

• nonn

Comments

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.

a(1) = 4 is the initial term of A075027: 4 is the smallest k such that d(k-1) < d(k) > d(k+1).

a(2) = 165 = A323379(1) is the smallest odd k such that d(k-1) < d(k) > d(k+1).

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).

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.

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).

Conjecture: no term is the product of more than 4 prime factors, counted with multiplicity.

Examples

			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.
.
                number of divisors of
                ======================  n-th   prime factorization
   n      a(n)  a(n)-1    a(n)  a(n)+1  prime        of a(n)
  --  --------  ------  ------  ------  -----  -------------------
   1         4       2       3       2      2     2 *  2
   2       165       6       8       4      3     3 *  5 * 11
   3      2275       8      12       6      5     5 *  5 *  7 * 13
   4     18473       8      12       8      7     7 *  7 * 13 * 29
   5     45617      10      12       8     11    11 * 11 * 13 * 29
   6     71383       8      12       8     13    13 * 17 * 17 * 19
   7    257393      10      12       8     17*   19 * 19 * 23 * 31
   8    257393      10      12       8     19    19 * 19 * 23 * 31
   9   1239907       8      16       6     23    23 * 31 * 37 * 47
  10   1275797       6      12       8     29    29 * 29 * 37 * 41
  11   1851847       8      12       8     31    31 * 31 * 41 * 47
  12   4411843       8      16      12     37    37 * 43 * 47 * 59
  13   6865337       8      12       8     41*   43 * 43 * 47 * 79
  14   6865337       8      12       8     43    43 * 43 * 47 * 79
  15   8312467       8      12       6     47    47 * 47 * 53 * 71
  16  15763207       8      12       8     53    53 * 59 * 71 * 71
  17  24157963      12      16       6     59    59 * 71 * 73 * 79
  18  33684317       6      16      12     61*   67 * 71 * 73 * 97
		

Crossrefs

Cf. A000005, A075027, A323379.