cp's OEIS Frontend

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.

%I A209203 #23 May 13 2013 01:50:04
%S A209203 6,12,16,28,34,36,54,76,78,84,114,124,132,138,142,148,154,166,168,208,
%T A209203 226,258,268,288,324,348,376,414,436,442,454,462,496,538,552,562,582,
%U A209203 588,684,714,736,744,798,804,814,832,882,894,912,946,972,994,1006
%N 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.
%C A209203 Numbers n such that n+25, 2n+125, and 3n+625 are prime.
%C A209203 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.
%C A209203 This sequence is infinite on Dickson's conjecture. [_Charles R Greathouse IV_, Mar 12 2012]
%H A209203 Sameen Ahmed Khan, <a href="/A209203/b209203.txt">Table of n, a(n) for n = 1..10000</a>
%H A209203 Sameen Ahmed Khan, <a href="http://arxiv.org/abs/1203.2083/">Primes in Geometric-Arithmetic Progression</a>, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).
%e A209203 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.
%t A209203 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
%t A209203 Select[Range[2,1100,2],And@@PrimeQ[{#+25,2#+125,3#+625}]&] (* _Harvey P. Dale_, Jan 06 2013 *)
%o A209203 (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
%Y A209203 Cf. A172367, A209202, A209204, A209205, A209206, A209207, A209208, A209209, A209210.
%K A209203 nonn
%O A209203 1,1
%A A209203 _Sameen Ahmed Khan_, Mar 06 2012