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

1,1

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.

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 *)

Crossrefs

Cf. A347113.

Sequence in context: A160080 A026247 A354236 * A226693 A306300 A105269

Adjacent sequences: A348776 A348777 A348778 * A348780 A348781 A348782

Keyword

nonn

Author

N. J. A. Sloane, Nov 13 2021

Status

approved