A171271 -id:A171271 - cp's OEIS frontend

A019434 Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.

3, 5, 17, 257, 65537

Offset: 1

It is conjectured that there are only 5 terms. Currently it has been shown that 2^(2^k) + 1 is composite for 5 <= k <= 32 (see Eric Weisstein's Fermat Primes link). - Dmitry Kamenetsky, Sep 28 2008
No Fermat prime is a Brazilian number. So Fermat primes belong to A220627. For proof see Proposition 3 page 36 in "Les nombres brésiliens" in Links. - Bernard Schott, Dec 29 2012
This sequence and A001220 are disjoint (see "Other theorems about Fermat numbers" in Wikipedia link). - Felix Fröhlich, Sep 07 2014
Numbers n > 1 such that n * 2^(n-2) divides (n-1)! + 2^(n-1). - Thomas Ordowski, Jan 15 2015
From Jaroslav Krizek, Mar 17 2016: (Start)
Primes p such that phi(p) = 2*phi(p-1); primes from A171271.
Primes p such that sigma(p-1) = 2p - 3.
Primes p such that sigma(p-1) = 2*sigma(p) - 5.
For n > 1, a(n) = primes p such that p = 4 * phi((p-1) / 2) + 1.
Subsequence of A256444 and A256439.
Conjectures:
1) primes p such that phi(p) = 2*phi(p-2).
2) primes p such that phi(p) = 2*phi(p-1) = 2*phi(p-2).
3) primes p such that p = sigma(phi(p-2)) + 2.
4) primes p such that phi(p-1) + 1 divides p + 1.
5) numbers n such that sigma(n-1) = 2*sigma(n) - 5. (End)
Odd primes p such that ratio of the form (the number of nonnegative m < p such that m^q == m (mod p))/(the number of nonnegative m < p such that -m^q == m (mod p)) is a divisor of p for all nonnegative q. - Juri-Stepan Gerasimov, Oct 13 2020
Numbers n such that tau(n)*(number of distinct ratio (the number of nonnegative m < n such that m^q == m (mod n))/(the number of nonnegative m < n such that -m^q == m (mod n))) for nonnegative q is equal to 4. - Juri-Stepan Gerasimov, Oct 22 2020
The numbers of primitive roots for the five known terms are 1, 2, 8, 128, 32768. - Gary W. Adamson, Jan 13 2022
Prime numbers such that every residue is either a primitive root or a quadratic residue. - Keith Backman, Jul 11 2022
If there are only 5 Fermat primes, then there are only 31 odd order groups which have a 2-group automorphism group. See the Miles Englezou link for a proof. - Miles Englezou, Mar 10 2025

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 137-141, 197.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see Table 1, p. 458.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.
Richard K. Guy, Unsolved Problems in Number Theory, A3.
Hardy and Wright, An Introduction to the Theory of Numbers, bottom of page 18 in the sixth edition, gives an heuristic argument that this sequence is finite.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 7, 70.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.

Anas AbuDaqa, Amjad Abu-Hassan, and Muhammad Imam, Taxonomy and Practical Evaluation of Primality Testing Algorithms, arXiv:2006.08444 [cs.CR], 2020.
Cyril Banderier, Pepin's Criterion For Fermat Numbers (in French)
Kent D. Boklan and John H. Conway, Expect at most one billionth of a new Fermat Prime!, arXiv:1605.01371 [math.NT], 2016.
P. Bruillard, S.-H. Ng, E. Rowell, and Z. Wang, On modular categories, arXiv:1310.7050 [math.QA], 2013-2015.
C. K. Caldwell, The Prime Glossary, Fermat number
Miles Englezou, Proof that 5 Fermat primes implies 31 odd order groups with 2-group automorphism groups
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
Salah Eddine Rihane, Chèfiath Awero Adegbindin, and Alain Togbé, Fermat Padovan And Perrin Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.2.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 65.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38. Local copy, included here with permission from the editors of Quadrature.
Yaryna Stelmakh, Homeomorphisms of the space of non-zero integers with the Kirch topology, arXiv:2101.04676 [math.GN], 2020.
Eric Weisstein's World of Mathematics, Fermat Number
Eric Weisstein's World of Mathematics, Fermat Prime
Eric Weisstein's World of Mathematics, Pepin's Test
Wikipedia, Fermat number
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A000215, A001146, A159611, A220627, A220570.
Subsequence of A147545 and of A334101. Cf. also A333788, A334092.
Cf. A045544.

[2^(2^n)+1 : n in [0..4] | IsPrime(2^(2^n)+1)]; // Arkadiusz Wesolowski, Jun 09 2011

Select[Table[2^(2^n) + 1, {n, 0, 4}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

for(i=0,10, isprime(2^2^i+1) && print1(2^2^i+1,", ")) \\ M. F. Hasler, Nov 21 2009

[2^(2^n)+1 for n in (0..4) if is_prime(2^(2^n)+1)] # G. C. Greubel, Mar 07 2019

a(n+1) = A180024(A049084(a(n))). - Reinhard Zumkeller, Aug 08 2010

a(n) = 1 + A001146(n-1), if 1 <= n <= 5. - Omar E. Pol, Jun 08 2018

A014224 Numbers k such that 3^k - 2 is prime.

Original entry on oeis.org

2, 4, 5, 6, 9, 22, 37, 41, 90, 102, 105, 317, 520, 541, 561, 648, 780, 786, 957, 1353, 2224, 2521, 6184, 7989, 8890, 19217, 20746, 31722, 37056, 69581, 195430, 225922, 506233, 761457, 1180181

Offset: 1

Jud McCranie

If n is of the form 4k + 3 then 3^n - 2 is composite, because 3^n - 2 = (3^4)^k*3^3 - 2 == 0 (mod 5). So there is no term of the form 4k + 3. If Q is a perfect number such that gcd(3(3^a(n) - 2), Q) = 1 then x = 3^(a(n) - 1)*(3^a(n) - 2)*Q is a solution of the equation sigma(x) = 3x + Q. See comment lines of the sequences A058959 and A171271. -  M. F. Hasler and Farideh Firoozbakht, Dec 07 2009
For all numbers n in this sequence, 3^(n-1)*(3^n-2) is a 2-hyperperfect number, cf. A007593, and no other 2-hyperperfect number seems to be known. - Farideh Firoozbakht and M. F. Hasler, Apr 25 2012
225922 is the last term in the sequence up to 500000. All n <= 500000 have been tested with the Miller-Rabin PRP test and/or PFGW. - Ryan Propper, Aug 18 2013
For n <= 506300 there is one additional term, 506233, a probable prime as tested by PFGW. - Ryan Propper, Sep 03 2013
a(35) > 10^6. - Ryan Propper, Jul 22 2015

Daniel Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem; Vol. 4, No. 2, Dec 1978, pp. 277-302. [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (pp. 114-134) [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing. [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]

Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010. See pp.79-80.
F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Henri and Renaud Lifchitz, PRP Records.

3^n - 2 = A058481(n).

Cf. A058959, A171271. - M. F. Hasler and Farideh Firoozbakht, Dec 07 2009

A014224 = {}; Do[If[PrimeQ[3^n - 2], Print[n]; AppendTo[A014224, n]], {n, 10^5}]; A014224 (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Do[If[Mod[n, 4] != 3 && PrimeQ[3^n - 2], Print[n]], {n, 10000}] (* M. F. Hasler and Farideh Firoozbakht, Dec 07 2009 *)

for(n=1,1e4,if(ispseudoprime(3^n-2),print1(n", "))) \\ Charles R Greathouse IV, Jul 19 2011

Corrected by Andrey V. Kulsha, Feb 04 2001
a(26) = 19217, a(27) = 20746 from Ryan Propper, May 11 2007
a(28) = 31722 from Henri Lifchitz, Oct 2002
a(29) = 37056 from Henri Lifchitz, Oct 2004
a(30) = 69581 from Henri Lifchitz, Jan 2005
a(31) = 195430 from Theodore Burton, Feb 2007
a(32) = 225922 from Ryan Propper, Aug 18 2013
a(33) = 506233 from Ryan Propper, Sep 02 2013
a(34) = 761457 from Ryan Propper, Jul 22 2015
a(35) = 1180181 from Jorge Coveiro, May 22 2020

A058959 Numbers k such that 3^k - 4 is prime.

Original entry on oeis.org

2, 3, 5, 21, 31, 37, 41, 53, 73, 101, 175, 203, 225, 455, 557, 651, 1333, 4823, 20367, 32555, 52057, 79371, 267267, 312155

Offset: 1

Robert G. Wilson v, Jan 13 2001

From M. F. Hasler and Farideh Firoozbakht, Oct 30 2009: (Start)
If Q is a perfect number such that gcd(Q, 3(3^a(n)-4))=1 then m=3^(a(n)-1) (3^a(n)-4)Q is a solution of the equation sigma(x)=3(x+Q). This is a result of the following theorem.
Theorem : If for a prime q, Q is a (q-1)-perfect number and p=q^k-q-1 is a prime such that gcd(Q, p*q)=1, then m=p*q^(k-1)*Q is a solution of the equation sigma(x)=q(x+Q). The proof is easy. (End)
From M. F. Hasler and Farideh Firoozbakht, Dec 07 2009: (Start)
2 is the only even term of this sequence because if n is an even number greater than 2 then 3^n-4=(3^(n/2)-2)*(3^(n/2)+2) is composite.
We have also found the following generalization of this theorem. See comment lines of the sequence A171271.
Theorem : If for a prime q, Q is a (q-1)-perfect number and for some integers k and m, p=q^k-m*q-1 is a prime such that gcd(Q, p*q)=1, then x=p*q^(k-1)*Q is a solution of the equation sigma(x)=q(x+m*Q). The proof is easy. (End)

Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Henri & Renaud Lifchitz, PRP Records.

Cf. A014224, A171271. - M. F. Hasler and Farideh Firoozbakht, Dec 07 2009

[n: n in [1..10^3]| IsPrime(3^n-4)]; // Vincenzo Librandi, Mar 24 2015

Do[ If[ PrimeQ[3^n - 4], Print[n] ], {n, 1, 3000} ]

for(n=1,10^3,if(ispseudoprime(3^n-4),print1(n,", "))) \\ Derek Orr, Mar 06 2015

a(18)=4823, corresponding to a certified prime, from Ryan Propper, Jun 30 2005
a(19)=20367 from Ray Chandler, Jul 25 2011
a(20)=32555, a(21)=52057 from Henri Lifchitz, Jan 2005
a(22)=79371 from Ray Chandler, Jul 25 2011
a(23)=267267 from Roman Ilyukhin, Oct 17 2014
a(24)=312155 from Roman Ilyukhin, Feb 28 2015

A050472 Numbers m such that 2*phi(m) = phi(m+1).

Original entry on oeis.org

2, 4, 16, 154, 256, 286, 364, 804, 1066, 2146, 3382, 4550, 6106, 7700, 8176, 9268, 11284, 12556, 12970, 16402, 19228, 19276, 20272, 25132, 26404, 27346, 29154, 29574, 35644, 36418, 38368, 39646, 40494, 47214, 52234, 54652, 65536, 84862

Offset: 1

Jud McCranie, Dec 24 1999

nonn

			phi(256)=128, phi(256+1)=2*128, so 256 is a member of the sequence.

R. K. Guy, Unsolved Problems Number Theory, Sect. B36.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Donovan Johnson)

Cf. A000010, A001274, A050473, A171271.

Select[Range[85000], 2EulerPhi[#]==EulerPhi[#+1] &] (* Stefano Spezia, Apr 07 2025 *)

isok(n) = 2*eulerphi(n) == eulerphi(n+1); \\ Michel Marcus, Aug 02 2018

Conjecture : a(n)/n^3 is bounded. Does lim n -> infinity a(n)/n^3 = 2 ? - Benoit Cloitre, Aug 07 2002

a(n) = A171271(n) - 1. - Ray Chandler, May 01 2015

A171262 Numbers n such that phi(n) = 2*phi(n+1).

Original entry on oeis.org

5, 13, 35, 37, 61, 73, 157, 193, 277, 313, 397, 421, 455, 457, 541, 613, 661, 665, 673, 733, 757, 877, 997, 1085, 1093, 1153, 1201, 1213, 1237, 1295, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2169, 2341, 2473, 2557, 2593, 2797, 2857

Offset: 1

Farideh Firoozbakht, Feb 23 2010

Theorem: A prime p is in the sequence iff 1/2*(p+1) is prime.
Proof: If both numbers p & 1/2*(p+1) are prime then phi(p)=p-1=2*(p-1)/2
2*(1/2*(p+1)-1)=2*phi(1/2*(p+1)), 1/2*(p+1) is odd so phi(1/2*(p+1))=
phi(p+1) hence phi(p)=2*phi(p+1), namely p is in the sequence.
Also if p is a prime term of the sequence
then phi(p)=2*phi(p+1) so
p-1=2*phi(p+1) or phi(p+1)=1/2*(p+1)-1 hence 1/2*(p+1)is prime.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A005383, A171271.

[n: n in [1..3*10^3] | EulerPhi(n) eq 2*EulerPhi(n+1)]; // Vincenzo Librandi, Apr 14 2015

Select[Range[2900],EulerPhi[ # ]==2EulerPhi[ #+1]&]

phi(35)=2*12=2*phi(35+1), so 35 is in the sequence.

A266276 a(n) is the smallest number k such that phi(k) = n*phi(k-1).

Original entry on oeis.org

2, 3, 7, 1261, 11242771

Offset: 1

Jaroslav Krizek, Jan 26 2016

a(n) >= A266269(n). - Max Alekseyev, Jan 26 2025

			a(3) = 7 because 7 is the smallest number k such that phi(k) = n*phi(k-1); phi(7) = 6 =3*phi(6) = 3*2.

Sequences of numbers n such that phi(n) = k*phi(n-1): {A001274 + 1} for k=1; A171271 = {A050472 + 1} for k=2; A266268 = {A067143 + 1} for k=3; A268126 = {A172314 + 1} for k=4; {A201253 + 1} for k=5.

Cf. A000010, A091439, A091456, A266269.

Magma
```
a:=func; [a(n):n in[1..5]];
```

a(n) = my(k=2, epk=1, enk); while ((enk=eulerphi(k)) != n*epk, epk = enk; k++); k; \\ Michel Marcus, Feb 20 2020

A266268 Numbers n such that phi(n) = 3*phi(n-1).

Original entry on oeis.org

7, 13, 19, 37, 73, 91, 97, 109, 163, 193, 433, 487, 577, 703, 769, 793, 925, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 4699, 5551, 6697, 7999, 8701, 10369, 10591, 11803, 12289, 16471, 17497, 18433, 33251, 39367, 52489, 56791, 79249, 124357, 127927, 137899

Offset: 1

Jaroslav Krizek, Dec 26 2015

nonn

Prime terms are in A058383.
See A266276(n) = the smallest numbers k such that phi(k) = n * phi(k-1) for n >=1: 2, 3, 7, 1261, 11242771, ...
Number of terms < 10^k: 1, 7, 17, 29, 41, 86, 205, 446, 1001, 2295, ..., . - Robert G. Wilson v, Jan 24 2016
All terms are == +-1 (mod 6) but mostly 1 (> 95%). - Robert G. Wilson v, Jan 24 2016

			19 is in the sequence because phi(19) = 18 = 3*phi(18) = 3*6.

Robert G. Wilson v, Table of n, a(n) for n = 1..2467 (first 405 terms from G. C. Greubel)

Cf. A000010, A058383, A171271 (numbers n such that phi(n) = 2*phi(n-1)), A266276.

[n: n in [2..2*10^5] | EulerPhi(n) eq 3*EulerPhi(n-1)]; // Vincenzo Librandi, Dec 26 2015

Select[Range[5000], EulerPhi[ # ]==3*EulerPhi[ #-1]&] (* G. C. Greubel, Dec 26 2015 *)

isok(n) = eulerphi(n) == 3*eulerphi(n-1); \\ Michel Marcus, Dec 27 2015

lista(nn) = for(n=1, nn, if(eulerphi(n) == 3*eulerphi(n-1), print1(n, ", "))); \\ Altug Alkan, Jan 24 2016

a(n) = A067143(n) + 1.

A258454 Numbers n such that phi(n) = 2*phi(n-2).

Original entry on oeis.org

3, 4, 5, 8, 17, 32, 257, 512, 527, 992, 1952, 2522, 5252, 6512, 7412, 10376, 23432, 23717, 26732, 27302, 35114, 36632, 37442, 45872, 47027, 49022, 51092, 65537, 78899, 84242, 92432, 98432, 98672, 114767, 115292, 131072, 227222, 231167, 240977, 328352, 369272

Offset: 1

Jaroslav Krizek, Sep 24 2015

nonn

Conjecture: a prime p is in the sequence iff p is a Fermat prime (A019434).

This is not correct: the first non-Fermat prime term is 83623937 = 2^18*11*29 + 1. - Joerg Arndt, Oct 11 2015

			phi(32) = 16 = 2*phi(30) = 2*8, so 32 is in the sequence.

Cf. A171271, A171262.

[n: n in [3..10000000] | EulerPhi(n) eq 2*EulerPhi(n-2)];

Select[Range@ 400000, EulerPhi@ # == 2 EulerPhi[# - 2] &] (* Michael De Vlieger, Sep 25 2015 *)

for(n=1, 1e6, if(eulerphi(n) == 2*eulerphi(n-2), print1(n", "))); \\ Altug Alkan, Sep 26 2015

A263811 Numbers k such that k = tau(k) * phi(k-1) + 1.

Original entry on oeis.org

3, 5, 17, 25, 49, 257, 289, 65537

Offset: 1

Jaroslav Krizek, Nov 04 2015

Numbers k such that k = A000005(k) * A000010(k-1) + 1.
The first 5 known Fermat primes from A019434 are in this sequence.
The next term, if it exists, must be greater than 2*10^7.
A prime p is in the sequence iff p is a Fermat prime (A019434) - see proof in A171271.
Observation: the known composite terms are squares of primes. - Omar E. Pol, Nov 04 2015
From Charlie Neder, Mar 02 2019: (Start)
Rearranging the definition gives (k-1)/phi(k-1) = tau(k), which means k-1 is in A007694. Since k-1 is thus 3-smooth, there are two possibilities:
1) k-1 is a power of 2 and tau(k) = 2, i.e., k is a Fermat prime,
2) k-1 is a 3-smooth number divisible by 6 and tau(k) = 3, i.e., k is a Pierpont number and the square of a prime.
In the second case, k-1 factors as (p-1)(p+1) for some p, and both parts are 3-smooth if and only if p is in {2,3,5,7,17} (2 and 3 are excluded since in those cases k-1 is not divisible by 6). Therefore, this sequence is complete if and only if there are no more Fermat primes. (End)

			17 is in this sequence because 17 = tau(17)*phi(16) + 1 = 2*8 + 1.

Cf. A000005, A000010, A019434.

Cf. A263810 (numbers k such that k = tau(k) * phi(k-2) + 1).

[n: n in [2..1000000] |  n eq NumberOfDivisors(n) * EulerPhi(n-1) + 1];

Select[Range[10^5], # == DivisorSigma[0, #] EulerPhi[# - 1] + 1 &] (* Michael De Vlieger, Nov 05 2015 *)

for(n=1, 1e5, if( n-1 == numdiv(n)*eulerphi(n-1) , print1(n, ", "))) \\ Altug Alkan, Nov 05 2015

A268126 Numbers n such that phi(n) = 4*phi(n-1).

Original entry on oeis.org

1261, 13651, 17557, 18721, 24511, 42121, 113611, 244531, 266071, 712081, 749911, 795691, 992251, 1080721, 1286731, 1458271, 1849471, 2271061, 2457691, 3295381, 3370771, 3414841, 3714751, 4061971, 4736491, 5314051, 5827081, 6566911, 6935083, 7303981, 7864081

Offset: 1

Jaroslav Krizek, Jan 26 2016

nonn

See A266276(n) = the smallest numbers k such that phi(k) = n * phi(k-1) for n >=1: 2, 3, 7, 1261, 11242771, ...

			1261 is in the sequence because phi(1261) = 1152 = 4*phi(1260) = 4*288.

Cf. A000010, A171271 (numbers n such that phi(n) = 2*phi(n-1)), A266268 (numbers n such that phi(n) = 3*phi(n-1)), A266276.

Cf. A256937.

[n: n in [2..10^7] | EulerPhi(n) eq 4*EulerPhi(n-1)]

Select[Range@10000000, EulerPhi@# == 4 EulerPhi[# - 1] &] (* Vincenzo Librandi, Jan 27 2016 *)

isok(n) = (eulerphi(n) == 4*eulerphi(n-1)); \\ Michel Marcus, Jan 27 2016

a(n) = A172314(n) + 1. - Michel Marcus, Jan 27 2016

cp's OEIS Frontend

A019434 Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.

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A014224 Numbers k such that 3^k - 2 is prime.

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A058959 Numbers k such that 3^k - 4 is prime.

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A050472 Numbers m such that 2*phi(m) = phi(m+1).

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A171262 Numbers n such that phi(n) = 2*phi(n+1).

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A266276 a(n) is the smallest number k such that phi(k) = n*phi(k-1).

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A266268 Numbers n such that phi(n) = 3*phi(n-1).

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