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 1-7 of 7 results.

A005537 Numbers m such that 4*3^m + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 885, 1005, 1254, 1635, 3306, 3522, 9602, 19785, 72698, 233583, 328689, 537918, 887535, 980925, 1154598, 1499606, 1936890, 2016951, 2143374
Offset: 1

Views

Author

N. J. A. Sloane, Chris K. Caldwell

Keywords

Comments

a(27) > 1.5*10^6. - Matthias Baur, Jan 16 2020
a(20) > 2*10^5. - Robert Price, Nov 23 2013
Primes resulting from a(1)-a(19) are confirmed primes (not probable primes) using BLS (N-1/N+1) test in pfgw. - Robert Price, Nov 23 2013
From Matthias Baur, Jan 16 2020: (Start)
Double checked to n=2*10^5, tested further to n=1.5*10^6 using the sieve programs newpgen and srsieve and using Jean Penné's LLR application (BLS (N-1/N+1) test).
a(20) was already known in 2005, but was not listed here until 2018 (see Prime Pages link). (End)
Because of the factorization 4*x^4 + 1 = (2*x^2 - 2*x + 1)*(2*x^2 + 2*x + 1), the only term divisible by 4 is 0. - Jeppe Stig Nielsen, Sep 12 2024

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005540, A125734.

Programs

  • Mathematica
    a[n_]:=If[PrimeQ[4*3^n + 1 ], n]; DeleteCases[Array[a, 40, 0], Null] (* Stefano Spezia, Nov 12 2018 *)
  • PARI
    is_a(m) = isprime(4*3^m + 1) \\ Michel Marcus, Jul 12 2013

Extensions

a(15)-a(17) from Douglas Burke (dburke(AT)nevada.edu)
a(18) from Mohammed Bouayoun (Mohammed.Bouayoun(AT)sanef.com), Jan 26 2004
a(19) from Robert Price, Nov 23 2013
a(20)-a(21) from Matthias Baur, Nov 07 2018
a(22) from Matthias Baur, Dec 06 2018
a(23)-a(24) from Matthias Baur, Jul 23 2019
a(25) from Matthias Baur, Dec 07 2019
a(26) from Matthias Baur, Jan 16 2020
a(27)-a(29) from Ryan Propper, May 08 2020

A205521 Numbers n such that 4*11^n - 1 is prime.

Original entry on oeis.org

1, 3, 7, 13, 25, 243, 349, 3645, 10245, 40647, 40749
Offset: 1

Views

Author

José María Grau Ribas, Jan 28 2012

Keywords

Crossrefs

Cf. A199020 (4*11^n-1).
Cf. A120378 (numbers n such that 2*11^n - 1 is prime).
Cf. A046865 (numbers n such that 4*5^n - 1 is prime).
Cf. A005540 (numbers n such that 4*3^n - 1 is prime).

Programs

  • Mathematica
    Table[If[PrimeQ[4*11^n-1],Print[n];n],{n,1,20000}]
  • PARI
    is(n)=ispseudoprime(4*11^n-1) \\ Charles R Greathouse IV, Jun 13 2017

Extensions

a(10)-a(11) from Dmitry Khomovsky, Aug 30 2017

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1
Offset: 2

Views

Author

Eric Chen, Jun 04 2018

Keywords

Comments

a(prime(j)) + 1 = A087139(j).
a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
a(251) > 73000, see A087139.

Links

Crossrefs

For the numbers k such that these forms are prime:
a1(b): numbers k such that (b-1)*b^k-1 is prime
a2(b): numbers k such that (b-1)*b^k+1 is prime
a3(b): numbers k such that (b+1)*b^k-1 is prime
a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
a5(b): numbers k such that b^k-(b-1) is prime
a6(b): numbers k such that b^k+(b-1) is prime
a7(b): numbers k such that b^k-(b+1) is prime
a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
.
b a1(b) a2(b) a3(b) a4(b) a5(b) a6(b) a7(b) a8(b)
--------------------------------------------------------------------
2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732
3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958
4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142
6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351
7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx
8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382
9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492
10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx
11 A046867 A057462 ------- ------- ------- ------- ------- -------
12 A079907 A251259 ------- ------- ------- A137654 ------- -------
13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
14 A273523 ------- ------- ------- ------- ------- ------- -------
15 ------- ------- ------- ------- ------- ------- ------- -------
16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).

Programs

  • PARI
    a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))

A136541 Numbers n such that sum of the proper divisors of n is equal to (3/4)*phi(n).

Original entry on oeis.org

33, 2889, 235953, 19129689
Offset: 1

Views

Author

Farideh Firoozbakht, Jan 08 2008

Keywords

Comments

If m>0 and p=4*3^m-1 is prime(m is in the sequence A005540) then n=3^m*p is in the sequence. Because sigma(n)-n=(1/2)*(3^(m+1)-1) *4*3^m-3^m*(4*3^m-1)=3^m*(2*3^m-1)=(3/4)*(2*3^(m-1))*((4*3^m-1)-1) =(3/4)*phi(3^m)*phi(p)=(3/4)*phi(3^m*p)=(3/4)*phi(n). The first four terms of the sequence are of such form if the 5th term is also of such form then it is equal to 823564514029689. Next term is greater than 2*10^9. Is it true that all terms are of the mentioned form?
a(5) > 10^12. - Giovanni Resta, Nov 03 2012

Examples

			sigma(33)-33=48-33=15=(3/4)*20=(3/4)*phi(33).

Crossrefs

Cf. A005540, A076373.

Programs

  • Mathematica
    Do[If[DivisorSigma[1,n]-n==3/4*EulerPhi@n,Print[n]],{n,2000000000}]

Formula

For n=1,2,3 & 4 a(n)=3^(2n-1)*(4*3^(2n-1)-1).

A205796 Numbers n such that 4*17^n - 1 is prime.

Original entry on oeis.org

1, 9, 33, 69, 427, 1009, 2667
Offset: 1

Views

Author

José María Grau Ribas, Jan 31 2012

Keywords

Crossrefs

Cf. A180431 (numbers n such that 4*17^n + 1 is prime).
Cf. A205521 (numbers n such that 4*11^n - 1 is prime).
Cf. A046865 (numbers n such that 4*5^n - 1 is prime).
Cf. A005540 (numbers n such that 4*3^n - 1 is prime).

Programs

  • Mathematica
    Table[If[PrimeQ[4*17^n-1],Print[n];n],{n,1,20000}]
  • PARI
    is(n)=ispseudoprime(4*17^n-1) \\ Charles R Greathouse IV, Jun 13 2017

A385115 Numbers k such that 2^4 * 3^k - 1 is prime.

Original entry on oeis.org

1, 3, 9, 13, 31, 43, 81, 121, 235, 1135, 1245, 1521, 2019, 2329, 3573, 11245, 15571, 37333, 54471, 70641
Offset: 1

Views

Author

Ken Clements, Aug 14 2025

Keywords

Comments

All terms are odd, since if k were even, N = 2^4 * 3^k would be a perfect square and N - 1 could be factored as the difference of squares, hence not prime.
a(21) > 10^5. - Michael S. Branicky, Aug 15 2025

Crossrefs

Cf. A005540, A005541, A387060.

Programs

  • Mathematica
    Select[Range[4000], PrimeQ[16 * 3^# - 1] &] (* Amiram Eldar, Aug 15 2025 *)
  • Python
    from gmpy2 import is_prime
print([k for k in range(1, 4_000, 2) if is_prime(16 * 3**k - 1)])

Extensions

a(17)-a(20) from Michael S. Branicky, Aug 15 2025

A387197 Numbers k such that 32 * 3^k - 1 is prime.

Original entry on oeis.org

0, 3, 4, 6, 46, 59, 84, 94, 124, 239, 267, 366, 371, 424, 616, 2139, 2299, 3523, 3563, 3843, 3923, 7627, 12751, 34798, 39911, 56568, 58779
Offset: 1

Views

Author

Ken Clements, Aug 21 2025

Keywords

Comments

a(28) > 10^5.
Conjecture: This sequence intersects with A387201 at k = 4 to form twin primes with center N = 2^5 * 3^4 = 2592 = A027856(10). Any such intersection has to be at an even k because if k is odd, either N-1 or N+1 has to be divisible by 5. A covering system can be constructed that eliminates all other intersections except where k = 4(mod 60), and for k > 4 with k = 4(mod 60), the search up to 10^5 makes the probability of another intersection in this residue class vanishingly small.

Links

Crossrefs

Cf. A027856, A003307, A005540, A005541, A385115, A387201.

Programs

  • Mathematica
    Select[Range[0, 4000], PrimeQ[32 * 3^# - 1] &] (* Amiram Eldar, Aug 21 2025 *)
  • Python
    from gmpy2 import is_prime
print([ k for k in range(4000) if is_prime(32 * 3**k - 1)])
Showing 1-7 of 7 results.

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

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