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A268480 Integers k such that A002110(k) is the average of two consecutive primes.

%I A268480 #38 May 19 2024 02:14:17
%S A268480 2,3,5,8,38,40,64,73,89,236,480,486
%N A268480 Integers k such that A002110(k) is the average of two consecutive primes.
%C A268480 In other words, the primorial numbers that are considered are those of the form (p + q)/2 where p and q are consecutive primes. Note that the initial values of (p - q)/2 are 1, 1, 1, 23, 239, 191, 331, 373, 1021.
%C A268480 A088256 is a subsequence of these primorials, which in turn are a subsequence of A024675.
%C A268480 Numbers k such that A038711(k) = A060270(k). - _Amiram Eldar_, May 19 2024
%H A268480 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>.
%e A268480 5 is a term because 2*3*5*7*11 = 2310 = (2309 + 2311)/2.
%e A268480 8 is a term because 2*3*5*7*11*13*17*19 = 9699690 = (9699667 + 9699713)/2.
%p A268480 P:= 2: count:= 0:
%p A268480 for n from 2 to 500 do
%p A268480   P:= P*ithprime(n);
%p A268480   # first try d=1
%p A268480   if isprime(P+1) then
%p A268480     good:= isprime(P-1);
%p A268480   elif isprime(P-1) then good:= false
%p A268480   else
%p A268480     for d from ithprime(n+1) by 2 do
%p A268480       if igcd(d,P) > 1 then next fi;
%p A268480       if isprime(P+d) then
%p A268480         good:= isprime(P-d); break
%p A268480       elif isprime(P-d) then
%p A268480         good:= false; break
%p A268480       fi
%p A268480     od;
%p A268480   fi;
%p A268480   if good then
%p A268480      count:= count+1;
%p A268480      A[count]:= n;
%p A268480   fi
%p A268480 od:
%p A268480 seq(A[i],i=1..count);  # _Robert Israel_, Aug 29 2016
%t A268480 prim[n_] := Times @@ Prime[Range[n]]; Select[Range[2, 100], Total[NextPrime[(p = prim[#]), {-1, 1}]] == 2*p &] (* _Amiram Eldar_, May 19 2024 *)
%o A268480 (PARI) a002110(n) = prod(k=1, n, prime(k));
%o A268480 for(n=2, 1e3, if((nextprime(a002110(n)) - a002110(n)) == (a002110(n) - precprime(a002110(n))), print1(n, ", ")))
%Y A268480 Cf. A002110, A024675, A038711, A060270, A088256.
%K A268480 nonn,more
%O A268480 1,1
%A A268480 _Altug Alkan_, Mar 21 2016