This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A330092 #5 Dec 04 2019 21:01:26
%S A330092 5,3,2,103,2437,6991,455033,252492571,8276659373,18749113741
%N A330092 The least prime that starts a chain of exactly n primes such that the product of each successive pair is a golden semiprime (A108540).
%C A330092 The question of the existence of arbitrary long chains of such primes was asked by _Jonathan Vos Post_ in A107768.
%C A330092 Such chains may be called "golden chains of primes". They are analogous to Cunningham chains: this sequence is analogous to A005602, as A108541 is analogous to A005384.
%e A330092 a(1) = 5 since 5 is not a lesser prime of a golden semiprime, i.e., it is not in A108541.
%e A330092 a(2) = 3 since 3 * 5 is a golden semiprime.
%e A330092 a(3) = 2 since {2, 3, 5} is a chain of 3 primes such that 2 * 3 and 3 * 5 are golden semiprimes.
%t A330092 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]];
%t A330092 goldChainLength[p_] := -1 + Length @ NestWhileList[goldPrime, p, # > 0 &];
%t A330092 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
%Y A330092 Cf. A005602, A107768, A108540, A108541.
%K A330092 nonn,more
%O A330092 1,1
%A A330092 _Amiram Eldar_, Dec 01 2019