A209203 Values of the difference d for 4 primes in geometric-arithmetic progression with the minimal sequence {5*5^j + j*d}, j = 0 to 3.

6, 12, 16, 28, 34, 36, 54, 76, 78, 84, 114, 124, 132, 138, 142, 148, 154, 166, 168, 208, 226, 258, 268, 288, 324, 348, 376, 414, 436, 442, 454, 462, 496, 538, 552, 562, 582, 588, 684, 714, 736, 744, 798, 804, 814, 832, 882, 894, 912, 946, 972, 994, 1006

Offset: 1

Author

Sameen Ahmed Khan, Mar 06 2012

Keywords

• nonn

Comments

Numbers n such that n+25, 2n+125, and 3n+625 are prime.

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 5.

This sequence is infinite on Dickson's conjecture. [Charles R Greathouse IV, Mar 12 2012]

Examples

			d = 12  then {5*5^j + j*d}, j = 0 to 3, is {5, 37, 149, 661}, which is 4 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, A209204, A209205, A209206, A209207, A209208, A209209, A209210.

Programs

Mathematica

p = 5; gapset4d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d}] == {True, True, True, True}, AppendTo[gapset4d, d]], {d, 0, 1000, 2}]; gapset4d
Select[Range[2,1100,2],And@@PrimeQ[{#+25,2#+125,3#+625}]&] (* Harvey P. Dale, Jan 06 2013 *)

PARI

forstep(n=2,1e3,[2,2,2,4],if(isprime(n+25)&&isprime(2*n+125)&&isprime(3*n+625),print1(n", "))) \\ Charles R Greathouse IV, Mar 12 2012