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

1,1

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.

Links

Table of n, a(n) for n=1..10.

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

Cf. A005602, A107768, A108540, A108541.

Sequence in context: A241182 A193799 A262225 * A064812 A282469 A216998

Adjacent sequences: A330089 A330090 A330091 * A330093 A330094 A330095

Keyword

nonn,more

Author

Amiram Eldar, Dec 01 2019

Status

approved