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A306395 Primes g such that 8*g + 2*p is a primorial for some twin prime p.

%I A306395 #25 Feb 27 2019 01:32:09
%S A306395 2,3,11,19,23,31,37,79,83,97,113,131,139,157,173,181,191,211,229,233,
%T A306395 239,241,251,263,271,281,293,331,337,359,367,379,419,431,439,449,503,
%U A306395 541,547,601,607,619,641,653,659,661,691,701,719,727,743,761,769,809
%N A306395 Primes g such that 8*g + 2*p is a primorial for some twin prime p.
%C A306395 So far, I find that there exists at least one prime g, and at least one twin prime p in A001097, such that 8g + 2p is a primorial.  Some of the related twin primes are rather large.  The twin related to a(112), for instance, is 242 digits long.  For each n, the program returns the primorial, g, g (mod 30) the twin prime (mod 30) and the twin prime.  These data are in a linked file.
%H A306395 Michael G. Kaarhus, <a href="/A306395/b306395.txt">Table of n, a(n) for n = 0..250</a>
%H A306395 Michael G. Kaarhus, <a href="/A306395/a306395.txt">Additional data</a>
%e A306395 n |  b# = 8 * g   +  2 * p     greater or lesser
%e A306395 --+----------------------------------------------
%e A306395 1 |  5# = 8 *  2  +  2 *    7  greater
%e A306395 2 |  5# = 8 *  3  +  2 *    3  lesser
%e A306395 3 |  7# = 8 * 11  +  2 *   61  greater
%e A306395 4 |  7# = 8 * 19  +  2 *   29  lesser
%e A306395 5 |  7# = 8 * 23  +  2 *   13  greater
%e A306395 6 | 11# = 8 * 31  +  2 * 1031  lesser
%o A306395 (CALC) #!/usr/bin/calc -q -f
%o A306395 global b=5, chck=list(), g=1, gt, mg30=2, mg6, mp30=7, n=1, oar=pfact(b)/2,
%o A306395 tpr=7, ts='greatr', fmt = "%4d%s%5d%s%7d%7d%9d%11s%s%d\n";
%o A306395 define bookem(an) {
%o A306395     mp30=mod(tpr, 30);
%o A306395     printf(fmt, n, '.', b, '#', an, mg30, mp30, ts, '  ', tpr);
%o A306395     n++; append(chck, an); return(an);
%o A306395 }
%o A306395 define incg() {
%o A306395     top: g=nextprime(g); mg6=mod(g, 6); mg30=mod(g, 30);
%o A306395     if (mg30 == 13 || mg30 == 17) {goto top;}
%o A306395     else {gt=g*4; return(mg30);}
%o A306395 }
%o A306395 define incb(p) {b=nextprime(p); oar=pfact(b)/2; return(b);}
%o A306395 print;
%o A306395 printf(fmt, 'n', '.', 'b', '#', 'g', 'g%30', 'twin%30', 'twin type', '  ', 'twin prime');
%o A306395 print '----------------------------------------------------------';
%o A306395 for (i=0; i<=1; i++) {g=nextprime(g); bookem(g); tpr=3; ts='lesser'; mg30=3;}
%o A306395 b=incb(b); while (g <= b) {incg();}
%o A306395 while (n <= 35) {
%o A306395     while (g > b) {
%o A306395         tpr=oar-gt;
%o A306395         if (tpr <= 7) {incb(b); continue;}
%o A306395         if (ptest(tpr, 200)) {
%o A306395             if (mg6 == 1 && ptest(tpr+2, 200)) {
%o A306395                     ts='lesser'; bookem(g); break;
%o A306395             }
%o A306395             else {if (ptest(tpr-2, 200)) {
%o A306395                     ts='greatr'; bookem(g); break;
%o A306395                 }
%o A306395             }
%o A306395         }
%o A306395         incb(b);
%o A306395     }
%o A306395     incg();
%o A306395     while (oar-gt > 0) {b=prevprime(b); oar=pfact(b)/2;}
%o A306395 }
%o A306395 print; chs=size(chck)-1; for (i=0; i <= chs; i++) {print i+1, chck[[i]];}
%Y A306395 Subsequence of A000040.  Supersequence of A218046.
%K A306395 nonn
%O A306395 1,1
%A A306395 _Michael G. Kaarhus_, Feb 12 2019