A107768 Integers p*q*r such that p*q and q*r are both golden semiprimes (A108540). Integers p*q*r such that p = A108541(j), q = A108542(j) = A108541(k) and r = A108542(k).

30, 1309, 50209, 299423, 4329769, 4661471, 13968601, 19867823, 49402237, 90419171, 95575609, 230236057, 289003081, 4195692049, 7752275351, 8857002097, 9759031489, 10956612769, 12930672109, 12991059409, 13494943703, 13807499677, 15195694009, 18253659551, 20769940297

Offset

1,1

Comments

Golden 3-almost primes.

Volumes of bricks (rectangular parallelepipeds) each of whose faces has golden semiprime area. How long a chain is possible of the form p(1) * p(2) * p(3) * ... * p(n) where each successive pair of values are factors of a golden semiprime? That is, if Zumkeller's golden semiprimes are the 2-dimensional case and the present sequence is the 3-dimensional case, is there a maximum n for an n-dimensional case?

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

30 = 2 * 3 * 5, where both 2*3=6 and 3*5=15 are golden semiprimes.
1309 = 7 * 11 * 17.
50209 = 23 * 37 * 59.

Mathematica

f[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]]; g[p_] := Module[{ p1 = f[p]}, If[p1 == 0, 0, p2 = f[p1]; If[p2 == 0, 0, p*p1*p2]]]; seq={}; p=1; Do[p = NextPrime[p]; gp = g[p]; If[gp > 0, AppendTo[seq, gp]], {300}]; seq (* Amiram Eldar, Nov 29 2019 *)

Crossrefs

Cf. A014612, A108540, A108541, A108542.

Sequence in context: A273416 A002456 A358163 * A353104 A048536 A369143

Adjacent sequences: A107765 A107766 A107767 * A107769 A107770 A107771

Keyword

easy,nonn

Author

Jonathan Vos Post, Jun 11 2005

Extensions

More terms from Amiram Eldar, Nov 29 2019

Status

approved