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%I #32 Dec 17 2013 06:20:11
%S 11,35,65,41,221,655,515,263,4265,893,4085,1031,3161,145,821,2083,
%T 2101,433,3743,2243,511,2623,5653,271,2885,4157,18023,9023,1151,4787,
%U 737,2141,2833,6181,3217,3635,715,4501,5381,4231,13265,823
%N Smallest k such that 6^n+k, 6^n+k+g, 6^n+k+2*g are consecutive primes in arithmetic progression.
%C The sequence starts at n=2 as there is no solution for n=1.
%C The primes are probable primes for n>23.
%H Pierre CAMI, <a href="/A233546/b233546.txt">Table of n, a(n) for n = 2..350</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression#Consecutive_primes_in_arithmetic_progression">Consecutive primes in arithmetic progression</a>
%e 6^2+11=47, 6^2+11+6=53, 6^2+11+2*6=59 and 47, 53, 59 are consecutive primes
%e and k=11 is minimal (since although 6^2+5=41, 6^2+5+6=47, 6^2+5+2*6=53 are primes, they are not consecutive primes), so a(2)=11. (Example clarified by _Jonathan Sondow_, Dec 16 2013.)
%t a[n_] := For[k = 1, True, k = k+2, p = 6^n+k; If[PrimeQ[p], q = NextPrime[p]; r = NextPrime[q]; g = q-p; If[g == r-q, Print["n = ", n, " k = ", k, " g = ", g, " ", {p, q, r}]; Return[k]]]]; Table[a[n], {n, 2, 100}] (* _Jean-François Alcover_, Dec 17 2013 *)
%o (PFGW & SCRIPT)
%o DIM n,1
%o DIM i
%o DIM J
%o DIM k
%o DIM pp
%o DIMS t
%o OPENFILEOUT myf,a(n).txt
%o LABEL a
%o SET n,n+1
%o IF n>350 THEN END
%o SET i,-1
%o SET j,0
%o SET k,0
%o LABEL b
%o SET i,i+2
%o SETS t,%d,%d,%d,%d\,;n;i;j;k
%o SET pp,6^n+i
%o PRP pp,t
%o IF ISPRP THEN GOTO c
%o GOTO b
%o LABEL c
%o SET j,j+2
%o SET pp,6^n+i+j
%o SETS t,%d,%d,%d,%d\,;n;i;j;k
%o PRP pp,t
%o IF ISPRP THEN GOTO d
%o GOTO c
%o LABEL d
%o IF j%6==0 THEN GOTO e
%o SET i,i+j
%o SET j,0
%o GOTO c
%o LABEL e
%o SET k,k+2
%o SETS t,%d,%d,%d,%d\,;n;i;j;k
%o SET pp,6^n+i+j+k
%o PRP pp,t
%o IF ISPRP && k==j THEN GOTO h
%o IF ISPRP THEN GOTO f
%o GOTO e
%o LABEL f
%o IF k%6==0 THEN GOTO g
%o SET i,i+j+k
%o SET j,0
%o SET k,0
%o GOTO c
%o LABEL g
%o SET i,i+j
%o SET j,k
%o SET k,0
%o GOTO e
%o LABEL h
%o WRITE myf,t
%o GOTO a
%Y Cf. A231576, A231578, A233550, A233742.
%K nonn
%O 2,1
%A _Pierre CAMI_, Dec 12 2013
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