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).

3, 5, 11, 17, 29, 59, 149, 179, 419, 1049, 2309, 9239, 11549, 25409, 180179, 270269, 300299, 330329, 390389, 420419, 4084079, 8678669, 106696589, 892371479, 2454021569, 3569485919, 4238764529, 4461857399, 4908043139, 6023507489

Offset: 1

Author

Pierre CAMI, Sep 29 2003

Keywords

• nonn

Comments

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.

Examples

			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.
The above prime is a(124). - _Robert G. Wilson v_, Jul 22 2015
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..104 (first 128 terms from Robert G. Wilson v) (shortened by _N. J. A. Sloane_, Jan 13 2019)

Crossrefs

Cf. A000040, A002110, A086916, A087700, A087731, A088676, A060735.

Programs

Mathematica

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 *)

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

Extensions

Edited by Jud McCranie, Oct 06 2003

Corrected by T. D. Noe, Nov 15 2006