OFFSET
1,1
COMMENTS
The question of the existence of arbitrary long chains of such primes was asked by Jonathan Vos Post in A107768.
EXAMPLE
a(1) = 5 since 5 is not a lesser prime of a golden semiprime, i.e., it is not in A108541.
a(2) = 3 since 3 * 5 is a golden semiprime.
a(3) = 2 since {2, 3, 5} is a chain of 3 primes such that 2 * 3 and 3 * 5 are golden semiprimes.
MATHEMATICA
goldPrime[p_] := Module[{x = GoldenRatio*p}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; q = If[x - p1 < p2 - x, p1, p2]; If[Abs[q - x] < 1, q, 0]];
goldChainLength[p_] := -1 + Length @ NestWhileList[goldPrime, p, # > 0 &];
max = 7; seq = Table[0, {max}]; count = 0; p = 1; While[count < max, p = NextPrime[p]; i = goldChainLength[p]; If[i <= max && seq[[i]] < 1, count++; seq[[i]] = p]]; seq
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Dec 01 2019
STATUS
approved