cp's OEIS Frontend

A087732 Smaller of twin primes of the form P=j*P(i)#-1 and P=j*P(i)#+1 with 0 < j < P(i+1), where P(i) denotes i-th prime and P(i)# the i-th primorial number A002110(i).

%I A087732 #24 Jan 13 2019 19:18:15
%S A087732 3,5,11,17,29,59,149,179,419,1049,2309,9239,11549,25409,180179,270269,
%T A087732 300299,330329,390389,420419,4084079,8678669,106696589,892371479,
%U A087732 2454021569,3569485919,4238764529,4461857399,4908043139,6023507489
%N A087732 Smaller of twin primes of the form P=j*P(i)#-1 and P=j*P(i)#+1 with 0 < j < P(i+1), where P(i) denotes i-th prime and P(i)# the i-th primorial number A002110(i).
%C A087732 Probably an infinite sequence. Using the UB874 program (UBASIC) I found the first 123 primes of the sequence for i <= 382. I think I have a proof that the sequence is infinite.
%H A087732 Charles R Greathouse IV, <a href="/A087732/b087732.txt">Table of n, a(n) for n = 1..104</a> (first 128 terms from Robert G. Wilson v) (shortened by _N. J. A. Sloane_, Jan 13 2019)
%e A087732 17=3*P(2)#-1 and 19=3*P(2)#+1 are twin primes, so 17 is in the sequence, corresponding to i=2, j=3. Again, 182*2633#-1 and 182*2633#+1 are prime twins, with j=182, i=382. These are 1111-digit twin primes.
%e A087732 The above prime is a(124). - _Robert G. Wilson v_, Jul 22 2015
%t A087732 f[n_] := Range[Prime[n + 1] - 1] Times @@ Prime@ Range@ n; s = Select[ Union@ Flatten@ Join[ Array[f, 10] - 1, Array[f, 11, 0] + 1], PrimeQ@# &]; s[[Select[ Range[-1 + Length@ s], s[[#]] + 2 == s[[# + 1]] &]]] (* _Robert G. Wilson v_, Jul 22 2015 *)
%o A087732 (PARI) do(lastprime)=my(v=List(),P=1,p=2); forprime(q=3,nextprime(lastprime\1+1), P*=p; for(j=1,q-1, if(isprime(j*P-1)&&isprime(j*P+1), listput(v, j*P-1))); p=q); Vec(v) \\ _Charles R Greathouse IV_, Jul 22 2015
%Y A087732 Cf. A000040, A002110, A086916, A087700, A087731, A088676, A060735.
%K A087732 nonn
%O A087732 1,1
%A A087732 _Pierre CAMI_, Sep 29 2003
%E A087732 Edited by _Jud McCranie_, Oct 06 2003
%E A087732 Corrected by _T. D. Noe_, Nov 15 2006