A209205 Values of the difference d for 6 primes in geometric-arithmetic progression with the minimal sequence {7*7^j + j*d}, j = 0 to 5.

144, 1494, 1740, 2040, 3324, 4044, 6420, 12804, 13260, 13464, 13620, 15444, 25824, 31524, 31674, 31680, 32124, 33720, 38064, 40410, 44634, 45804, 46260, 51810, 54510, 56100, 58914, 60810, 68004, 69114, 70794, 74574, 76050, 77694, 80580, 81510, 82434, 89244

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 multiple of 3# = 6 and coprime to 7.

Examples

			d = 1494 then {7*7^j + j*d}, j = 0 to 5, is {7, 1543, 3331, 6883, 22783, 125119}, which is 6 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, A209202, A209203, A209204, A209206, A209207, A209208, A209209, A209210.

Programs

Mathematica

p = 7; gapset6d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d, p*p^5 + 5*d}] == {True, True, True, True, True, True}, AppendTo[gapset6d, d]], {d, 0, 100000, 2}]; gapset6d