A348779 Primes in A347113 in order of appearance.

5, 2, 3, 13, 7, 11, 17, 29, 19, 23, 41, 43, 53, 47, 31, 73, 37, 59, 61, 101, 71, 67, 109, 83, 139, 89, 79, 103, 107, 113, 149, 137, 131, 127, 181, 97, 151, 163, 167, 233, 173, 179, 191, 193, 197, 281, 157, 293, 223, 227, 239, 241, 251, 199, 257, 263, 349, 269, 283, 277, 401, 409, 311, 421, 211, 313

Offset: 1

Author

N. J. A. Sloane, Nov 13 2021

Keywords

• nonn

Comments

From Michael De Vlieger, Nov 13 2021: (Start)

Let s = A347113, j = s(n-1)+1 and k = s(n). Prime k|j = q such that j/q = p, p < q, both primes, in all cases except the first 3, i.e., s(7), s(8), and s(11), with (j, k) = {(95, 5), (6, 2), (15, 3)} respectively.

In other words, squarefree semiprime j = pq, p < q, yields k = q outside of the first 3 primes in s. Are all prime s(n), n > 219 in this category?

Prime k implies k | j, since k = j is not permitted in s, k < j.

There is 1 instance of composite k | j, i.e., s(33) = 25, with j = 75. Are there any others?

The reverse relation j to k is that j is the product of at least one prime divisor p | k and at least one prime q that does not divide k. When k is prime p, j = pq.

Contains local minima in s aside from s(1). A consequence of forbidden j = k in s is that local minima are nonadjacent.

(End)

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..23082
• Michael De Vlieger, Log-log scatterplot of A347113(n) for n=1..2^12 showing primes in this sequence in green. Maxima in A347113 appear in red and local minima in blue.

Crossrefs

Cf. A347113.

Programs

Mathematica

c[] = 0; m = 1; k = 2; Reap[Monitor[Do[If[IntegerQ@ Log2[i], While[c[k] > 0, k++]]; Set[n, k]; While[Or[c[n] > 0, n == m + 1, GCD[n, m + 1] == 1], n++]; If[PrimeQ[n], Sow[n]]; Set[c[n], i]; m = n, {i, 648}], i]][[-1, -1]] (* _Michael De Vlieger, Nov 13 2021 *)