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A027570 Initial members of prime decaplets (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26, p+30, p+32).

%I A027570 #47 Nov 04 2021 13:10:15
%S A027570 9853497737,21956291867,22741837817,164444511587,179590045487,
%T A027570 217999764107,231255798857,242360943257,666413245007,696391309697,
%U A027570 867132039857,974275568237,976136848847,1002263588297,1086344116367
%N A027570 Initial members of prime decaplets (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26, p+30, p+32).
%C A027570 All terms are congruent to 167 (modulo 210). - _Matt C. Anderson_, May 29 2015
%H A027570 Matt C. Anderson and Dana Jacobsen, <a href="/A027570/b027570.txt">Table of n, a(n) for n = 1..10000</a> [first 100 terms from Matt C. Anderson]
%H A027570 Tony Forbes and Norman Luhn, <a href="http://www.pzktupel.de/ktuplets.htm">Prime k-tuplets</a>
%H A027570 Norman Luhn, <a href="http://www.pzktupel.de/SMArchiv/10tup2.zip">1 million terms of A027570</a>, zip compressed (7.94 MB) (2021).
%p A027570 a := 1:
%p A027570 for b to 25 do
%p A027570 a := a*ithprime(b):
%p A027570 end do:
%p A027570 a;
%p A027570 # now 'a' is the product of the primes less than 100.
%p A027570 composite_small := proc (n::integer)
%p A027570 description "procedure to determine if n has a prime factor less than 100";
%p A027570 if igcd(2305567963945518424753102147331756070, n) = 1 then return false
%p A027570 else return true;
%p A027570 end if;
%p A027570 end proc:
%p A027570 # so composite_small tests if there are any factors 2 through 97.
%p A027570 #begin initialization section
%p A027570 p := [0, 2, 6, 12, 14, 20, 24, 26, 30, 32];
%p A027570 o := [7517, 10247, 12137, 14447, 14867, 17177, 21377, 24107, 25997, 28727];
%p A027570 m := 30030;
%p A027570 #end initialization section
%p A027570 # implement isprime(m*n+o+p)
%p A027570 with(ArrayTools):
%p A027570 os:=Size(o,2):
%p A027570 ps:=Size(p,2):
%p A027570 #here ps is 10 so a prime constellation of length 10.
%p A027570 loopstop := 10^11:
%p A027570 loopstart := 0:
%p A027570 for n from loopstart to loopstop do
%p A027570 for a to os do
%p A027570 counter := 0; wc := 0; wd := 0;
%p A027570   while `and`(wd > -10, wd < ps) do
%p A027570   wd := wd+1;
%p A027570   if composite_small(m*n+o[a]+p[wd]) = false then wd := wd+1
%p A027570   else wd := -10 end if;
%p A027570   end do;
%p A027570 if wd >= 9 then
%p A027570 while `and`(counter >= 0, wc < ps) do
%p A027570   wc := wc+1;
%p A027570   if isprime(m*n+o[a]+p[wc]) then counter := counter+1;
%p A027570   else counter := -1
%p A027570   end if;
%p A027570 end do;
%p A027570 end if;
%p A027570 if counter = ps then print(m*n+o[a]) end if;
%p A027570 end do:
%p A027570 end do:
%p A027570 # _Matt C. Anderson_, Apr 15 2015
%o A027570 (Perl) use ntheory ":all"; say for sieve_prime_cluster(1,1e13, 2,6,12,14,20,24,26,30,32); # _Dana Jacobsen_, Sep 30 2015
%Y A027570 Cf. A027569, A202361, A202362.
%K A027570 nonn
%O A027570 1,1
%A A027570 _Warut Roonguthai_