login
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).
0
5, 3, 2, 103, 2437, 6991, 455033, 252492571, 8276659373, 18749113741
OFFSET
1,1
COMMENTS
The question of the existence of arbitrary long chains of such primes was asked by Jonathan Vos Post in A107768.
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.
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