A107768 - OEIS

%I #13 Dec 13 2022 11:01:28

%S 30,1309,50209,299423,4329769,4661471,13968601,19867823,49402237,

%T 90419171,95575609,230236057,289003081,4195692049,7752275351,

%U 8857002097,9759031489,10956612769,12930672109,12991059409,13494943703,13807499677,15195694009,18253659551,20769940297

%N 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).

%C Golden 3-almost primes.

%C 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?

%H Amiram Eldar, <a href="/A107768/b107768.txt">Table of n, a(n) for n = 1..10000</a>

%e 30 = 2 * 3 * 5, where both 2*3=6 and 3*5=15 are golden semiprimes.

%e 1309 = 7 * 11 * 17.

%e 50209 = 23 * 37 * 59.

%t 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 *)

%Y Cf. A014612, A108540, A108541, A108542.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Jun 11 2005

%E More terms from _Amiram Eldar_, Nov 29 2019