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.

User: Rashid Naimi

Rashid Naimi's wiki page.

Rashid Naimi has authored 4 sequences.

A339174 Let b(1) = 2 and let b(n+1) be the least prime expressible as k*(b(n)-1)*b(n)+1; this sequence gives the values of k in order.

Original entry on oeis.org

1, 1, 1, 2, 5, 9, 6, 79, 16, 219, 580, 387, 189, 7067, 1803, 6582, 31917, 18888, 20973, 132755, 11419, 50111
Offset: 1

Author

Rashid Naimi, Nov 25 2020

Keywords

Comments

The corresponding primes in order are 3, 7, 43, 3613, 65250781, P17, P34, P70, P141, P284, P571, P1144, P2290, P4584, P9170, P18344, P36692, P73387, P146778, P293560, P587124, P1174253.
After each iteration the number of decimal digits is roughly twice that of the previous iteration. These primes can generally be easily certified using the N-1 method since all the prime factors for N-1 are known.

Examples

			[Corrected by _Peter Munn_, Nov 05 2022]
For p = 2, the smallest k for which f(k) = k*(p-1)*p+1 is prime is 1 because we have: f(1) = k*(p-1)*p+1 = 1*(2-1)*2+1 = 3.
This sets p = 3 for the next iteration for which the smallest k for which f(k) is prime is 1: f(1) = k*(p-1)*p+1 = 1*(3-1)*3+1 = 7.
		

Crossrefs

Programs

  • PARI
    my(p=2, k=1); while(1, my(runningP=k*(p-1)*p+1); if(ispseudoprime(runningP), print1(k, ", "); k=1; p=runningP; , k=k+1))
    
  • PARI
    my(k=[1, 1, 1, 2, 5, 9, 6, 79, 16, 219, 580, 387, 189, 7067, 1803, 6582, 31917, 18888, 20973, 132755, 11419, 50111], p=2); for(i=1, #k, p=k[i]*(p-1)*p+1); print("\n", p, "\n"); \\ to produce the P587124 prime
    
  • Python
    from sympy import isprime
    A339174_list, a = [2], 2
    while len(A339174_list) < 10:
        k, c, b = 1, 1, (a-1)*a
        while True:
            c += b
            if isprime(c):
                A339174_list.append(k)
                a = c
                break
            k += 1 # Chai Wah Wu, Dec 04 2020

Formula

Nested f(k) = k*(p-1)*p+1 for p=2. After each iteration the last obtained f(k) is substituted for p. The primes can be certified using OpenPFGW by adding each previous iteration to the helper file.

Extensions

a(22) from Rashid Naimi, Jan 13 2023

A306601 Let b(1) = 3 and let b(n+1) be the least prime expressible as k*(b(n)-1)*b(n)-1; this sequence gives the values of k in order.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 5, 360, 142, 104, 34, 1904, 3127, 253, 1219, 8755, 16222, 7672, 22515
Offset: 1

Author

Rashid Naimi, Apr 10 2019

Keywords

Comments

The corresponding primes in order are 5, 19, 683, 1863223, P14, P29, P57, P117, P235, P472, P945, P1893, P3789, P7581, P15164, P30332, P60668, P121339, P242682.
After each iteration the number of decimal digits is roughly twice that of the previous iteration. These primes can generally be easily certified using the N+1 method since all the prime factors for N+1 are known.

Examples

			For p = 3, the smallest k for which f(k) = k*(p-1)*p-1 is prime is 1:
f(1) = k*(p-1)*p-1 = 1*(3-1)*3-1 = 5.
This sets p = 5 for the next iteration for which the smallest k for which f(k) is prime is 1:
f(1) = k*(p-1)*p-1 = 1*(5-1)*5-1 = 19.
This sets p = 19 for the next iteration for which the smallest k for which f(k) is prime is 2:
f(2) = k*(p-1)*p-1 = 2*(19-1)*19-1 = 683.
This sets p = 683 for the next iteration for which the smallest k for which f(k) is prime is 4:
f(4) = k*(p-1)*p-1 = 4*(683-1)*683-1 = 1863223.
This sets p = 1863223 for the next iteration for which the smallest k for which f(k) is prime is 8:
f(8) = k*(p-1)*p-1 = 8*(1863223-1)*1863223-1 = P14.
		

Crossrefs

Cf. A000058.

Programs

  • PARI
    p=3; k=1; while(1, runningP=k*(p-1)*p-1; if(ispseudoprime(runningP), print1(k,", "); k=1; p=runningP;, k=k+1))
    
  • PARI
    /* The largest prime (P242682) can be generated by using the code: */ k=[1, 1, 2, 4, 8, 16, 5, 360, 142, 104, 34, 1904, 3127, 253, 1219, 8755, 16222, 7672, 22515]; p=3; for(i=1, #k, p=k[i]*(p-1)*p-1); print("\n", p, "\n")

Formula

Nested f(k) = k*(p-1)*p-1 for p=3. After each iteration the last obtained f(k) is substituted for p. The primes can be certified using OpenPFGW by adding each previous iteration to the helper file.

Extensions

Definition clarified by Charlie Neder, Jun 03 2019
a(17) from Rashid Naimi, Aug 23 2019
a(18) from Rashid Naimi, Oct 22 2019
a(19) from Rashid Naimi, Aug 01 2020

A319224 Integers q for which f(q) = ((((q - 2)! - 1) / q) - 1) / (q + 1) is a prime number.

Original entry on oeis.org

7, 11, 19, 61, 2557
Offset: 1

Author

Rashid Naimi, Sep 13 2018

Keywords

Comments

For q < 7, f(q) is not an integer.
f(q) for q = 2557 is a PRP7592.
According to Wilson's theorem, f(q) can be an integer only if q is prime.
a(6) > 30000. - Michael S. Branicky, Apr 18 2025

Crossrefs

Programs

  • Mathematica
    a[q_]:=If[PrimeQ[((((q - 2)! - 1) / q) - 1) / (q + 1)], q]; DeleteCases[Array[a, 100], Null] (* Stefano Spezia, Nov 04 2018 *)
  • PARI
    forprime(q=7, 2557, my(p = ((((q - 2)! - 1) / q) - 1) / (q + 1)); if(ispseudoprime(p), print1(q, ", ")))

A319304 Integers q for which f(q) = ((((q - 1)! + 1) / q) + 1) / (q + 1) is a prime number.

Original entry on oeis.org

7, 17, 31, 67, 89
Offset: 1

Author

Rashid Naimi, Sep 16 2018

Keywords

Comments

f(89) is a 131-digit prime.
According to Wilson's theorem, f(q) can be an integer only if q is prime.
a(6) > 30000. - Michael S. Branicky, Apr 19 2025

Crossrefs

Programs

  • Magma
    [n: n in  [1..100] | IsPrime(n) and IsPrime((((Factorial(n-1)+1) div n)+1) div (n+1))]; // Vincenzo Librandi, Sep 21 2018
  • Mathematica
    Select[Prime[Range[100]], PrimeQ[((((# - 1)! + 1) / #) + 1) / (# + 1)] &] (* Vincenzo Librandi, Sep 21 2018 *)
  • PARI
    forprime(q=7, 89, my(p = ((((q - 1)! + 1) / q) + 1) / (q + 1)); if(ispseudoprime(p), print1(q, ", ")))
    

Extensions

1621 and 1699, which do not belong here, removed by Rashid Naimi, Mar 21 2019