A209202 Values of the difference d for 3 primes in geometric-arithmetic progression with the minimal sequence {3*3^j + j*d}, j = 0 to 2.

2, 8, 10, 20, 22, 28, 38, 50, 52, 62, 70, 92, 98, 100, 118, 122, 128, 140, 142, 170, 202, 218, 220, 230, 232, 248, 260, 268, 272, 302, 308, 328, 350, 358, 380, 392, 400, 430, 440, 470, 478, 482, 512, 532, 538, 548, 562, 568, 598, 632, 638, 650, 700, 710, 730

Offset: 1

Author

Sameen Ahmed Khan, Mar 06 2012

Keywords

• nonn

Comments

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j. Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be even and coprime to 3.

Examples

			d = 8 then {3*3^j + j*d}, j = 0 to 2, is  {3, 17, 43}, which is 3 primes in geometric-arithmetic progression.
		

Links

• Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
• Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Crossrefs

Cf. A172367, A209203, A209204, A209205, A209206, A209207, A209208, A209209, A209210.

Programs

Mathematica

p = 3; gapset3d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d}] == {True, True, True}, AppendTo[gapset3d, d]], {d, 0, 1000, 2}]; gapset3d