login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A171381 Numbers n such that (3^n + 1)/2 is prime. 3
1, 2, 4, 16, 32, 64 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Note that n must be a power of 2 (cf. A138083).

Similar to Fermat primes (A019434), and for the same reasons we expect this sequence to be finite as well.

The numbers (3^n + 1)/2 are strong-probable-primes to base 3, so don't test with that base. - Don Reble, Jun 15 2010

From Paul Bourdelais, Oct 13 2010: (Start)

Terms in sequence factored to 1E18:

(3^(2^21)+1)/2 has factors: 155189249

(3^(2^22)+1)/2 is composite: RES64: [A158D7ED3E1CC427] (425462 sec)

(3^(2^23)+1)/2 is composite: RES64: [B0F07A3D55C5082A] (3424080 sec)

(3^(2^26)+1)/2 has factors: 3221225473

(3^(2^28)+1)/2 has factors: 12348030977

(3^(2^29)+1)/2 has factors: 77309411329

(3^(2^31)+1)/2 has factors: 4638564679681

(3^(2^32)+1)/2 has factors: 206158430209

(3^(2^34)+1)/2 has factors: 50474455662593

(3^(2^36)+1)/2 has factors: 911220261519361

(3^(2^38)+1)/2 has factors: 6597069766657

(3^(2^39)+1)/2 has factors: 46179488366593

(3^(2^44)+1)/2 has factors: 15586676835352577

(3^(2^45)+1)/2 has factors: 16044073672507393

(3^(2^49)+1)/2 has factors: 7881299347898369

(3^(2^51)+1)/2 has factors: 891712726219358209

(3^(2^54)+1)/2 has factors: 180143985094819841

For all other n<55 (specifically, n=24, 25, 27, 30, 33, 35, 37, 40, 41, 42, 43, 46, 47, 48, 50, 52, 53), no factor < 10^18 has been found.

(End)

REFERENCES

Anders Bjorn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), pp. 441-446.

LINKS

Table of n, a(n) for n=1..6.

I. J. Calvo, A note on factors of generalized Fermats numbers, Applied Math. Letters 13, (2000), pp.1-5. [Gives divisibility criteria for 3^(2^m)+1 by primes of the form p=3*2^n+1 (p=7, 13, 97, 193 ...) (Theorem 2.1) and for primes of this form when they are divisors of Fermat numbers (Theorem 2.2)]

M. F. Hasler and G. Guninski, Eliminating some further terms

W. Keller, Factors of generalized Fermat numbers found after Bjorn & Riesel

W. Keller, Prime factors of generalized Fermat numbers F'_m(3) = F_m(3) / 2 and complete factoring status

EXAMPLE

(3^(2^0)+1)/2 = (3^1+1)/2 = 2 is prime

(3^(2^1)+1)/2 = (3^2+1)/2 = 5 is prime

(3^(2^2)+1)/2 = (3^4+1)/2 = 41 is prime

(3^(2^3)+1)/2 = (3^8+1)/2 = 3281 is divisible by 17=1+2^4.

(3^(2^4)+1)/2 = (3^16+1)/2 = 21523361 is prime

(3^(2^5)+1)/2 = (3^32+1)/2 = 926510094425921 is prime

(3^(2^6)+1)/2 = (3^64+1)/2 = 1716841910146256242328924544641 is prime

(3^(2^7)+1)/2 = (3^128+1)/2 is divisible by 257=1+2^8, so 2^7 is not a member.

(3^(2^8)+1)/2 = (3^256+1)/2 is divisible by 1+2^9*24, so 2^8 is not a member.

(3^(2^15)+1)/2 is divisible by 2^(2^4)+1, so 2^15 is not a member. - Georgi Guninski, Jun 13 2010

(3^(2^19)+1)/2 is divisible by 13631489, so 2^19 is not a member. - Paul Zimmermann, Jun 14 2010

(3^(2^20)+1)/2 is 5-composite so 2^20 is not a member. - Serge Batalov, Jun 14 2010

According to PFGW, 2^20 is not in the sequence: PFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14] (3^1048576+1)/2 is composite: RES64: [9EE4CA1AABB9A816] (3229 sec) Base 5. Base 3 is useless here [cf. comment by Don Reble - Ed.]. - Georgi Guninski, Jun 15 2010

2^23 is not in the sequence (listed as "Composite but no factor known" on the second Keller link). - Serge Batalov, Jun 18 2010.

Verified 2^23 yields a composite: base 2 (PFGWv3.3.1). - Paul Bourdelais, Apr 25 2011.

PROG

(MAGMA) IsPrime((3^(2^15)+1) div 2); [From Jon E. Schoenfield, Jun 13 2010, shows that 15 is not a member]

(PARI) is_A171381(n)=ispseudoprime(3^n\2+1)  \\ - M. F. Hasler, Oct 02 2012

CROSSREFS

Cf. A019434, A093625 (the primes), A138083 (exponents of powers of 2).

Sequence in context: A173746 A095803 A032464 * A274497 A145119 A081411

Adjacent sequences:  A171378 A171379 A171380 * A171382 A171383 A171384

KEYWORD

nonn,hard

AUTHOR

Joao Carlos Leandro da Silva (zxawyh66(AT)yahoo.com), Dec 07 2009

EXTENSIONS

Edited by N. J. A. Sloane, Dec 09 2009

Incorrect terms a(7)-a(15) deleted Jun 12 2010 by Jon E. Schoenfield

The next term, if it exists, is at least 2^19. - Georgi Guninski, Jun 13 2010

A comment in A093625 from Don Reble (djr(AT)nk.ca), Apr 28 2004, says the next term, if it exists, is >= 2^21.

n=2^21 yields a number divisible by 1+2^22*37. - M. F. Hasler, Jun 14 2010

Edited by N. J. A. Sloane, Jun 12 2010 - Jun 16 2010

Edited by M. F. Hasler, Oct 02 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 7 05:39 EST 2016. Contains 278841 sequences.