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

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

Author

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

Keywords

• nonn
• more

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

• C. K. Caldwell, The Prime Pages
• Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, On Perfect Totient Numbers, J. Integer Sequences, 6 (2003), #03.4.5.
• P. Loomis, M. Plytage and J. Polhill, Summing up the Euler 'phi' function, The College Mathematics Journal, vol. 39 (2008), pp. 34-42.
• H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*3^n+1 and 2*3^n-1, Math. Comp., 26 (1972), 995-998.

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