cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A088676 Values of "i" in A087732 and the index of the prime in A087731.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 9, 9, 9, 9, 9, 9, 9, 10, 11, 12, 12, 14, 15, 16, 17, 17, 17, 18, 20, 20, 20, 20, 22, 22, 23, 24, 25, 25, 26, 27, 28, 30, 31, 33, 34, 36, 36, 38, 41, 44, 49, 53, 55, 56, 59, 61, 62, 63, 65, 69, 69, 78, 85, 87, 99
Offset: 1

Views

Author

Jud McCranie, Oct 04 2003

Keywords

Examples

			17 = 3*P(2)#-1 and 19 = 3*P(2)#+1 are twin primes, so 2 is in the sequence (the 4th term).

Crossrefs

Cf. A002110, A087732.

Extensions

More terms from Ray Chandler, Oct 05 2003

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).

Original entry on oeis.org

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

Views

Author

Pierre CAMI, Sep 29 2003

Keywords

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

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

A086916 Number of digits in terms of A087732.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 9, 9, 10, 10, 10, 10, 10, 10, 12, 13, 14, 15, 18, 20, 21, 23, 23, 23, 25, 28, 28, 29, 29, 32, 33, 35, 37, 38, 39, 41, 43, 45, 47, 50, 55, 57, 62, 62, 66, 73, 79, 92, 101, 106, 108, 115, 121, 123, 126, 131, 140, 141
Offset: 1

Views

Author

Pierre CAMI, Sep 25 2003

Keywords

Comments

Number of digits of the twin primes of the form j*P(i)# - 1 and j*P(i)# + 1 with 0 < j < P(i+1), P(i)= i rank primes, P(i)# = primorial of P(i) (A002110).

Examples

			a(23)=9 since A087732(23)=106696589 has 9 digits.
a(31)=12 since A087732(31)=103515091679 has 12 digits.

Crossrefs

Cf. A086916, A087651, A087730, A087731, A087732, A088676.

Extensions

Edited by Ray Chandler, Oct 05 2003

A087651 Sequence of primorials P# (cf. A002110) such that j*P# has twin prime neighbors for some j with 0 < j < prime following P.

Original entry on oeis.org

2, 6, 6, 6, 30, 30, 30, 30, 210, 210, 2310, 2310, 2310, 2310, 30030, 30030, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 223092870, 223092870, 223092870, 223092870, 223092870, 223092870, 223092870, 6469693230, 200560490130
Offset: 0

Views

Author

Pierre CAMI, Sep 28 2003

Keywords

Comments

I think I have found a proof that the sequence is infinite.

Examples

			I have found the first 122 values of this sequence, 121 and 122 are 2557#, 2557# with j=303 and j=2307, 1087 and 1088 digits.

Crossrefs

The j values are in A087730, P values are in A087731, i values are in A088676.
Cf. A002110, A086916, A087732.
Smaller of twin primes are in A087732. Number of digits in twin primes are in A086916.

Extensions

Edited by Jud McCranie and Ray Chandler, Oct 05 2003
Showing 1-4 of 4 results.

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