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%I #5 Dec 04 2019 21:01:26
%S 5,3,2,103,2437,6991,455033,252492571,8276659373,18749113741
%N 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 The question of the existence of arbitrary long chains of such primes was asked by _Jonathan Vos Post_ in A107768.
%C 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 a(1) = 5 since 5 is not a lesser prime of a golden semiprime, i.e., it is not in A108541.
%e a(2) = 3 since 3 * 5 is a golden semiprime.
%e a(3) = 2 since {2, 3, 5} is a chain of 3 primes such that 2 * 3 and 3 * 5 are golden semiprimes.
%t 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 goldChainLength[p_] := -1 + Length @ NestWhileList[goldPrime, p, # > 0 &];
%t 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 Cf. A005602, A107768, A108540, A108541.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Dec 01 2019
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