A306395 - OEIS

%I #25 Feb 27 2019 01:32:09

%S 2,3,11,19,23,31,37,79,83,97,113,131,139,157,173,181,191,211,229,233,

%T 239,241,251,263,271,281,293,331,337,359,367,379,419,431,439,449,503,

%U 541,547,601,607,619,641,653,659,661,691,701,719,727,743,761,769,809

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

%C 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 Michael G. Kaarhus, <a href="/A306395/b306395.txt">Table of n, a(n) for n = 0..250</a>

%H Michael G. Kaarhus, <a href="/A306395/a306395.txt">Additional data</a>

%e n |  b# = 8 * g   +  2 * p     greater or lesser

%e --+----------------------------------------------

%e 1 |  5# = 8 *  2  +  2 *    7  greater

%e 2 |  5# = 8 *  3  +  2 *    3  lesser

%e 3 |  7# = 8 * 11  +  2 *   61  greater

%e 4 |  7# = 8 * 19  +  2 *   29  lesser

%e 5 |  7# = 8 * 23  +  2 *   13  greater

%e 6 | 11# = 8 * 31  +  2 * 1031  lesser

%o (CALC) #!/usr/bin/calc -q -f

%o global b=5, chck=list(), g=1, gt, mg30=2, mg6, mp30=7, n=1, oar=pfact(b)/2,

%o tpr=7, ts='greatr', fmt = "%4d%s%5d%s%7d%7d%9d%11s%s%d\n";

%o define bookem(an) {

%o     mp30=mod(tpr, 30);

%o     printf(fmt, n, '.', b, '#', an, mg30, mp30, ts, '  ', tpr);

%o     n++; append(chck, an); return(an);

%o }

%o define incg() {

%o     top: g=nextprime(g); mg6=mod(g, 6); mg30=mod(g, 30);

%o     if (mg30 == 13 || mg30 == 17) {goto top;}

%o     else {gt=g*4; return(mg30);}

%o }

%o define incb(p) {b=nextprime(p); oar=pfact(b)/2; return(b);}

%o print;

%o printf(fmt, 'n', '.', 'b', '#', 'g', 'g%30', 'twin%30', 'twin type', '  ', 'twin prime');

%o print '----------------------------------------------------------';

%o for (i=0; i<=1; i++) {g=nextprime(g); bookem(g); tpr=3; ts='lesser'; mg30=3;}

%o b=incb(b); while (g <= b) {incg();}

%o while (n <= 35) {

%o     while (g > b) {

%o         tpr=oar-gt;

%o         if (tpr <= 7) {incb(b); continue;}

%o         if (ptest(tpr, 200)) {

%o             if (mg6 == 1 && ptest(tpr+2, 200)) {

%o                     ts='lesser'; bookem(g); break;

%o             }

%o             else {if (ptest(tpr-2, 200)) {

%o                     ts='greatr'; bookem(g); break;

%o                 }

%o             }

%o         }

%o         incb(b);

%o     }

%o     incg();

%o     while (oar-gt > 0) {b=prevprime(b); oar=pfact(b)/2;}

%o }

%o print; chs=size(chck)-1; for (i=0; i <= chs; i++) {print i+1, chck[[i]];}

%Y Subsequence of A000040.  Supersequence of A218046.

%K nonn

%O 1,1

%A _Michael G. Kaarhus_, Feb 12 2019