A128421 -id:A128421 - cp's OEIS frontend

A128420 Numbers n such that p(n)# + p(n+1)# -1 is prime, where p(n)# is the product of first n primes (A002110).

Original entry on oeis.org

0, 1, 3, 6, 8, 10, 12, 37, 72, 92, 142, 295, 1529, 1625, 1914, 2276, 4423

Offset: 1

Pierre CAMI, Mar 02 2007

nonn

Cf. A002110, A057704, A128421.

Corrected by Emeric Deutsch, Mar 06 2007

Edited by Ray Chandler, Mar 13 2007

A128657 Numbers k such that p(k+1)# - p(k)# - 1 is prime where p(i)# = product of first i primes = A002110(i).

Original entry on oeis.org

1, 2, 3, 4, 8, 24, 35, 56, 79, 132, 281, 342, 631, 1350, 4929, 5952, 8128

Offset: 1

Pierre CAMI, Mar 19 2007

Cf. A002110, A128420, A128421, A128658, A128659, A128660, A128661, A128662.

isok(n) = {my(pp=prod(k=1, n, prime(k))); isprime(pp*prime(n+1) - pp -1);} \\ Michel Marcus, Sep 20 2019

a(17) from Michael S. Branicky, Oct 06 2024

A128658 Numbers k such that p(k+1)# - p(k)# + 1 is prime where p(i)# = product of first i primes = A002110(i).

Original entry on oeis.org

1, 3, 7, 14, 80, 224, 280, 287, 652, 1509, 2952, 3915, 7771

Offset: 1

Pierre CAMI, Mar 19 2007

Cf. A002110, A128420, A128421, A128657, A128659, A128660, A128661, A128662.

isok(n) = {my(pp=prod(k=1, n, prime(k))); isprime(pp*prime(n+1) - pp + 1);} \\ Michel Marcus, Sep 20 2019

a(13) from Michael S. Branicky, Oct 06 2024

A128659 Numbers k such that p(k+1)# - p(k)# - p(k-1)# - 1 is prime, where p(i)# = product of first i primes = A002110(i).

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 9, 13, 14, 20, 30, 40, 44, 86, 101, 135, 152, 171, 205, 4404, 12227

Offset: 1

Pierre CAMI, Mar 19 2007

			1 is a term since A002110(2) - A002110(1) - A002110(0) - 1 = 6 - 2 - 1 - 1 = 2.

Cf. A002110, A128420, A128421, A128657, A128658, A128660, A128661, A128662.

{1}~Join~Flatten[Position[Partition[Rest[FoldList[Times,1,Prime[Range[210]]]],3,1],?(PrimeQ[#[[3]]-#[[2]]-#[[1]]-1]&),{1},Heads->False]]+1 (* This generates the first 18 terms of the sequence.  To generate the 19th term, change the Range constant to 4410, but it will take a very long time to run. *) (* _Harvey P. Dale, Apr 23 2014 *)

isok(k) = ispseudoprime(vecprod(primes(k+1)) - vecprod(primes(k)) - vecprod(primes(k-1)) - 1); \\ Michel Marcus, May 07 2025

a(1) = 1 inserted by Michael S. Branicky, May 07 2025

a(21) from Michael S. Branicky, May 10 2025

A128660 Numbers n such that p(n+1)#-p(n)#-p(n-1)#+1 is prime where p(i)#=product of first i primes=A002110(i).

Original entry on oeis.org

2, 6, 14, 15, 23, 31, 97, 144, 983, 2587, 3163, 6689

Offset: 1

Pierre CAMI, Mar 19 2007

nonn

Cf. A002110, A128420, A128421, A128657, A128658, A128659, A128661, A128662.

Flatten[Position[#[[3]]-#[[2]]-#[[1]]+1&/@Partition[FoldList[Times,1,Prime[ Range[ 6700]]] ,3,1],?PrimeQ]] (* _Harvey P. Dale, Jul 25 2013 *)

A128661 Numbers k such that p(k+1)# + p(k)# - p(k-1)# - 1 is prime where p(i)# = product of first i primes = A002110(i).

Original entry on oeis.org

3, 7, 14, 15, 21, 32, 34, 47, 67, 69, 174, 369, 568, 3193, 3882, 5426

Offset: 1

Pierre CAMI, Mar 19 2007

Cf. A002110, A128420, A128421, A128657, A128658, A128659, A128660, A128662.

Flatten[Position[Partition[Rest[FoldList[Times,1,Prime[Range[ 600]]]],3,1], ?(PrimeQ[#[[3]]+#[[2]]-#[[1]]-1]&),{1},Heads-> False]]+1 (* _Harvey P. Dale, May 26 2013 *)

a(14)-a(16) from Michael S. Branicky, Jul 22 2023

A128662 Numbers k such that p(k+1)# + p(k)# - p(k-1)# + 1 is prime where p(i)# = product of first i primes = A002110(i).

Original entry on oeis.org

8, 13, 14, 15, 30, 41, 71, 85, 97, 132, 221, 282, 424, 509, 1346, 1348, 3684, 8224

Offset: 1

Pierre CAMI, Mar 19 2007

Cf. A002110, A128420, A128421, A128657, A128658, A128659, A128660, A128661.

Flatten[Position[Partition[FoldList[Times,Prime[Range[1350]]],3,1],?(PrimeQ[ #[[2]]+#[[3]]-#[[1]]+1]&),1,Heads->False]]+1 (* _Harvey P. Dale, May 30 2018 *)

P(n) = factorback(primes(n)); \\ A002110
isok(k) = ispseudoprime(P(k+1)+P(k)-P(k-1)+1); \\ Michel Marcus, Sep 11 2022

a(17) from Michael S. Branicky, Jun 23 2023

a(11) corrected and a(18) from Michael S. Branicky, Dec 11 2024

cp's OEIS Frontend

A128420 Numbers n such that p(n)# + p(n+1)# -1 is prime, where p(n)# is the product of first n primes (A002110).

Views

Author

Keywords

Crossrefs

Extensions

A128657 Numbers k such that p(k+1)# - p(k)# - 1 is prime where p(i)# = product of first i primes = A002110(i).

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

A128658 Numbers k such that p(k+1)# - p(k)# + 1 is prime where p(i)# = product of first i primes = A002110(i).

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

A128659 Numbers k such that p(k+1)# - p(k)# - p(k-1)# - 1 is prime, where p(i)# = product of first i primes = A002110(i).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A128660 Numbers n such that p(n+1)#-p(n)#-p(n-1)#+1 is prime where p(i)#=product of first i primes=A002110(i).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A128661 Numbers k such that p(k+1)# + p(k)# - p(k-1)# - 1 is prime where p(i)# = product of first i primes = A002110(i).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A128662 Numbers k such that p(k+1)# + p(k)# - p(k-1)# + 1 is prime where p(i)# = product of first i primes = A002110(i).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions