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).

5, 3, 2, 103, 2437, 6991, 455033, 252492571, 8276659373, 18749113741

Offset: 1

Author

Amiram Eldar, Dec 01 2019

Keywords

• nonn
• more

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.

Examples

			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.
		

Crossrefs

Cf. A005602, A107768, A108540, A108541.

Programs

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