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A348779 Primes in A347113 in order of appearance. 6
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 (list; graph; refs; listen; history; text; internal format)
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, 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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 13 2021
STATUS
approved

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Last modified February 23 11:40 EST 2024. Contains 370283 sequences. (Running on oeis4.)