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A227856 Sequence of pairs k,g with k<3*2^n the smallest such that 3*2^n+k, 3*2^n+k+g, 3*2^n+k+2*g are three consecutive primes in arithmetic progression starting at n=5 as there is not any solution for n<5.

%I A227856 #13 Jun 02 2025 08:36:23
%S A227856 55,6,7,12,173,6,173,6,205,6,229,6,113,6,203,6,95,6,475,6,163,6,119,
%T A227856 12,377,18,1045,6,133,12,551,24,131,12,259,6,1105,42,539,6,1487,18,
%U A227856 1295,12,5,12,289,36,311,36,269,6,2833,6,1813,18,835,6,319,6,587,6,239,30,1225,6,1825,12,973,12,89,30,551,12,1805,30,1039,18,1219,6
%N A227856 Sequence of pairs k,g with k<3*2^n the smallest such that 3*2^n+k, 3*2^n+k+g, 3*2^n+k+2*g are three consecutive primes in arithmetic progression starting at n=5 as there is not any solution for n<5.
%C A227856 The ratio k/n^2 is in average near 0.8 and < 7 for n<701.
%C A227856 The ratio g/n^2 is in average near 0.5 and < 4 for n<701.
%C A227856 If 3*2^n+k > 10^22 the numbers are probable primes.
%H A227856 Pierre CAMI, <a href="/A227856/b227856.txt">Table of n, a(n) for n = 5..1396</a>
%e A227856 3*2^5+55=151, 3*2^5+55+6=157 3*2^5+55*2*6=163
%e A227856 151, 157, 163 three consecutive primes in arithmetic progression 6, so first pair is 55, 6
%o A227856 PFGW & SCRIPTIFY
%o A227856 SCRIPT
%o A227856 DIM i
%o A227856 DIM j
%o A227856 DIM k
%o A227856 DIM n,4
%o A227856 DIM pp
%o A227856 DIM qq
%o A227856 DIM rr
%o A227856 DIMS t
%o A227856 OPENFILEOUT myf,a(n).txt
%o A227856 LABEL loop1
%o A227856 SET n,n+1
%o A227856 SET i,-1
%o A227856 SET j,0
%o A227856 SET k,0
%o A227856 LABEL a
%o A227856 SET i,i+2
%o A227856 SETS t,%d,%d\,;n;i
%o A227856 SET pp,3*2^n+i
%o A227856 PRP pp,t
%o A227856 IF ISPRP THEN GOTO b
%o A227856 GOTO a
%o A227856 LABEL b
%o A227856 SET j,j+2
%o A227856 SETS t,%d,%d,%d\,;n;i;j
%o A227856 SET qq,pp+j
%o A227856 PRP qq,t
%o A227856 IF ISPRP THEN GOTO c
%o A227856 GOTO b
%o A227856 LABEL c
%o A227856 SET k,k+2
%o A227856 SETS t,%d,%d,%d,%d\,;n;i;j;k
%o A227856 SET rr,qq+k
%o A227856 PRP rr,t
%o A227856 IF ISPRP THEN GOTO d
%o A227856 GOTO c
%o A227856 LABEL d
%o A227856 IF j==k THEN GOTO x
%o A227856 SET i,i+j
%o A227856 SET pp,qq
%o A227856 SET j,0
%o A227856 SET k,0
%o A227856 GOTO b
%o A227856 LABEL x
%o A227856 WRITE myf,t
%o A227856 GOTO loop1
%Y A227856 Cf. A230699, A230852
%K A227856 nonn
%O A227856 5,1
%A A227856 _Pierre CAMI_, Nov 01 2013