A209210 Values of the difference d for 11 primes in geometric-arithmetic progression with the minimal sequence {11*11^j + j*d}, j = 0 to 10.

443687580, 591655080, 1313813550, 2868131100, 3525848580, 3598823970, 4453413120, 6075076800, 6644124480, 7429693770, 9399746580, 11801410530, 12450590250

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 5# = 30 and coprime to 11.

Examples

			d = 591655080 then {11*11^j + j*d}, j = 0 to 10, is {11, 591655201, 1183311491, 1774979881, 2366781371, 2960046961, 3569417651, 4355944441, 7091188331, 31262320321, 291228221411}, which is 11 primes in geometric-arithmetic progression.
		

Links

• Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Crossrefs

Cf. A172367, A209202, A209203, A209204, A209205, A209206, A209207, A209208, A209209.

Programs

Mathematica

p = 11; gapset11d = {}; 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, p*p^6 + 6*d, p*p^7 + 7*d, p*p^8 + 8*d, p*p^9 + 9*d, p*p^10 + 10*d}] == {True, True, True, True, True, True, True, True, True, True, True}, AppendTo[gapset11d, d]], {d, 0, 10^8, 2}]