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%I #19 Nov 22 2021 10:06:57
%S 84,114,138,168,258,324,348,462,552,588,684,714,744,798,882,894,972,
%T 1176,1602,1734,2196,2256,2442,2478,2568,2646,2658,2688,3036,3162,
%U 3444,3906,4524,5154,5406,5544,5766,5796,6018,6456,6594,6636,6936,7272,7938,8736
%N Values of the difference d for 5 primes in geometric-arithmetic progression with the minimal sequence {5*5^j + j*d}, j = 0 to 4.
%C 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 5.
%C Subsequence of A209203. - _Zak Seidov_, Jul 06 2013
%H Sameen Ahmed Khan, <a href="/A209204/b209204.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 d = 114 then {5*5^j + j*d}, j = 0 to 4, is {5, 139, 353, 967, 3581}, which is 5 primes in geometric-arithmetic progression.
%t p = 5; gapset5d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d}] == {True, True, True, True, True}, AppendTo[gapset5d, d]], {d, 2, 10000, 2}]; gapset5d
%Y Cf. A172367, A209202, A209203, A209205, A209206, A209207, A209208, A209209, A209210.
%K nonn
%O 1,1
%A _Sameen Ahmed Khan_, Mar 06 2012
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