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 1-10 of 14 results. Next

A100857 a(n) = greatest prime factor of (prime(n) - 1)! + prime(n).

Original entry on oeis.org

3, 5, 29, 727, 3628811, 6599, 6385271, 1291883, 315389197, 123662437024088191, 2107707330310062524424576119, 50717732086848522419, 14602944428087660261, 55353784249983832172137, 16166897314580017
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 20 2004

Keywords

Examples

			n=6: (prime(6)-1)! + prime(6) = 12! + 13 = 479001600 + 13 = 479001613 = 29*2503*6599, therefore a(6) = 6599.

Links

Crossrefs

Cf. A006530, A100856, A100858.

Programs

  • Mathematica
    FactorInteger[(#-1)!+#][[-1,1]]&/@Prime[Range[15]] (* Harvey P. Dale, Jul 23 2011 *)

Formula

a(n) = A006530(A100856(n)).
a(n) > A000040(n).

A073309 Primes of the form k! + k + 1.

Original entry on oeis.org

2, 3, 5, 29, 727, 3628811, 80658175170943878571660636856403766975289505440883277824000000000053
Offset: 1

Views

Author

Rick L. Shepherd, Jul 24 2002

Keywords

Comments

a(6) = 3628811 and a(7), a 68-digit number, have been certified prime with Primo.

Examples

			a(4) = 6! + 6 + 1 = 727, a prime, so 727 is in this sequence (6 = A073308(4)).

Crossrefs

Cf. A073308 (corresponding n), A100858.

Programs

  • Mathematica
    f[n_]:=n!+n+1; lst={};Do[p=f[n];If[PrimeQ[p],AppendTo[lst,p]],{n,0,2*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 02 2009 *)
  • PARI
    for(n=0,1960,p=n!+n+1; if(isprime(p),print1(p,",")))

Formula

a(n) = A073308(n)! + A073308(n) + 1.

A100595 Numbers k such that (prime(k)-1)! + prime(k)^9 is prime.

Original entry on oeis.org

9, 10, 17, 137
Offset: 1

Views

Author

Jonathan Vos Post, Nov 30 2004

Keywords

Comments

There are no more such k up to k=150. Computed in collaboration with Ray Chandler.
a(5) > 1000. - Jinyuan Wang, Apr 11 2020
a(5) > 2700. - Michael S. Branicky, Jul 03 2024

Examples

			a(1) = 9 because (prime(9)-1)! + prime(9)^9 = (23-1)! + 23^9 = 1124000729578760341463 is the smallest prime of this form.
a(2) = 10 because (prime(10)-1)! + prime(10)^9 = (29-1)! + 29^9 = 304888344611713875008649975869 is the 2nd smallest prime of this form.
a(3) = 17, but prime(17) = 59 yields a number that would take 2 full lines of this page; and a(4) = 137 because prime(137) = 773 yields a prime of this form which is 1975 digits long. Note also that 773 = prime(137) = prime(prime(34)).

Crossrefs

Cf. A100596, A100598, A100858.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^9], AppendTo[lst, n]], {n, 12^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
  • PARI
    is(k) = ispseudoprime((prime(k)-1)! + prime(k)^9); \\ Jinyuan Wang, Apr 11 2020

Formula

Numbers k such that (prime(k)-1)! + prime(k)^9 is prime, where prime(k) is the k-th prime.

A100598 Numbers k such that (prime(k)-1)! + prime(k)^8 is prime.

Original entry on oeis.org

1, 2, 4, 5, 8, 10
Offset: 1

Views

Author

Jonathan Vos Post, Nov 30 2004

Keywords

Comments

k = {1, 2, 4, 5, 8, 10} yields primes p(k) = {2, 3, 7, 11, 19, 29}. There are no more such k up to k=100. Computed in collaboration with Ray Chandler.
a(7) > 600. - Jinyuan Wang, Apr 10 2020
a(7) > 900. - Tyler NeSmith, May 05 2021
a(7) > 2500. - Michael S. Branicky, Jul 03 2024

Examples

			a(1) = 1 because (prime(1)-1)! + prime(1)^8 = (2-1)! + 2^8 = 257 is the smallest prime of that form.
a(5) = 8 because (prime(8)-1)! + prime(8)^8 = (19-1)! + 19^8 = 304888344611713861001750412961 is the 5th smallest prime of that form.

Crossrefs

Cf. A100595, A100599, A100858.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^8], AppendTo[lst, n]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
  • PARI
    is(k) = ispseudoprime((prime(k)-1)! + prime(k)^8); \\ Jinyuan Wang, Apr 10 2020

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

A100599 Numbers k such that (prime(k)-1)! + prime(k)^7 is prime.

Original entry on oeis.org

7, 14, 16, 59
Offset: 1

Views

Author

Jonathan Vos Post, Nov 30 2004

Keywords

Comments

k = {7, 14, 16, 59} yields primes p(k) = {17, 43, 53, 277}. There are no more such k up to k=100. Computed in collaboration with Ray Chandler.
a(5) > 600. - Jinyuan Wang, Apr 10 2020
a(5) > 2500. - Michael S. Branicky, Jul 02 2024

Examples

			a(1) = 7 because (prime(7)-1)! + prime(7)^7 = (17-1)! + 17^7 = 20923200226673 is the smallest prime of that form.

Crossrefs

Cf. A100598, A100600, A100858.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^7], AppendTo[lst, n]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
  • PARI
    is(k) = ispseudoprime((prime(k)-1)! + prime(k)^7); \\ Jinyuan Wang, Apr 10 2020

A100600 Numbers k such that (prime(k)-1)! + prime(k)^6 is prime.

Original entry on oeis.org

3, 4, 29, 32, 133
Offset: 1

Views

Author

Jonathan Vos Post, Nov 30 2004

Keywords

Comments

k = {3, 4, 29, 32, 133} yields primes p(n) = {5, 7, 109, 131, 751}. There are no more such k up to k=100. Computed in collaboration with Ray Chandler.
a(6) > 600. - Jinyuan Wang, Apr 10 2020
a(6) > 2500. - Michael S. Branicky, Jul 02 2024

Examples

			a(1) = 3 because (prime(3)-1)! + prime(3)^6 = (5-1)! + 5^6 = 15649 is the smallest prime of that form.

Crossrefs

Cf. A100599, A100602, A100858.

Programs

Formula

Numbers k such that (prime(k)-1)! + prime(k)^6 is prime, where prime(k) is the k-th prime.

Extensions

a(5) from Jinyuan Wang, Apr 10 2020

A100605 Numbers k such that (prime(k)-1)! + prime(k)^2 is prime.

Original entry on oeis.org

1, 2, 4, 10, 26
Offset: 1

Views

Author

Jonathan Vos Post, Nov 30 2004

Keywords

Comments

k = {1, 2, 4, 10, 26} yields primes p(k) = {2, 3, 7, 29, 101}. There are no more such k up to k=100. Verified by Ray Chandler.
a(6) > 750. - Jinyuan Wang, Apr 10 2020
a(5) > 2500. - Michael S. Branicky, Jul 02 2024

Examples

			a(3) = 4 because (prime(4)-1)! + prime(4)^2 = (7-1)! + 7^2 = 720 + 49 = 769 is prime and is the 3rd such prime of that form.

Crossrefs

Cf. A100604, A100858.

Programs

  • PARI
    is(k) = ispseudoprime((prime(k)-1)! + prime(k)^2); \\ Jinyuan Wang, Apr 10 2020

Formula

Numbers k such that (prime(k)-1)! + prime(k)^2 is prime, where prime(k) is the k-th prime.

A100856 a(n) = (prime(n) - 1)! + prime(n).

Original entry on oeis.org

3, 5, 29, 727, 3628811, 479001613, 20922789888017, 6402373705728019, 1124000727777607680023, 304888344611713860501504000029, 265252859812191058636308480000031
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 20 2004

Keywords

Crossrefs

Cf. A100857, A100858, A000040, A000142.

A100596 Numbers k such that (prime(k)-1)! + prime(k)^10 is prime.

Original entry on oeis.org

2, 8, 15, 33, 52, 205, 751
Offset: 1

Views

Author

Jonathan Vos Post, Nov 30 2004

Keywords

Comments

k = {2, 8, 15, 33, 52, 205} yields primes p(k) = {3, 19, 47, 137, 239, 1259}. There are no more such k up to k=150. Computed in collaboration with Ray Chandler.
a(7) > 600. - Jinyuan Wang, Apr 10 2020
a(8) > 2700. - Michael S. Branicky, Jul 03 2024

Examples

			a(1) = 2 because (prime(2)-1)! + prime(2)^10 = (3-1)! + 3^10 = 59051 is the smallest prime of that form.
a(2) = 8 because (prime(8)-1)! + prime(8)^10 = (19-1)! + 19^10 = 6408504771985801 is the 2nd smallest prime of that form.

Crossrefs

Cf. A100595, A100858.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^10], AppendTo[lst, n]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
Select[Range[250],PrimeQ[(Prime[#]-1)!+Prime[#]^10]&] (* The program generates the first 6 terms of the sequence. *) (* Harvey P. Dale, Dec 27 2024 *)
  • Python
    from math import factorial
from sympy import isprime, prime
def afind(limit, startat=1):
    for k in range(startat, limit+1):
        s = str(k)
        pk = prime(k)
        if isprime( factorial(pk-1) + pk**10 ):
            print(k, end=", ")
afind(100) # Michael S. Branicky, Nov 30 2021

Formula

Primes of the form (prime(k)-1)! + prime(k)^10, where prime(k) is the k-th prime.

Extensions

a(6) from Jinyuan Wang, Apr 10 2020
a(7) from Michael S. Branicky, Nov 30 2021

A100602 Numbers k such that (prime(k)-1)! + prime(k)^5 is prime.

Original entry on oeis.org

6, 8, 11, 17, 2286
Offset: 1

Views

Author

Jonathan Vos Post, Nov 30 2004

Keywords

Comments

k = {6, 8, 11, 17} yields primes p(k) = {13, 19, 31, 59}. There are no more such k up to k=100. Computed in collaboration with Ray Chandler.
a(5) > 600. - Jinyuan Wang, Apr 10 2020

Examples

			a(1) = 6 because (prime(6)-1)! + prime(6)^5 = (13-1)! + 13^5 = 479372893 is the first prime of this form.

Crossrefs

Cf. A100600, A100858.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^5], AppendTo[lst, n]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
  • PARI
    is(k) = ispseudoprime((prime(k)-1)! + prime(k)^5); \\ Jinyuan Wang, Apr 10 2020

Formula

Numbers k such that (prime(k)-1)! + prime(k)^5 is prime, where prime(k) is the k-th prime.

Extensions

a(5) from Michael S. Branicky, Jul 02 2024
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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