A333788 -id:A333788 - 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

A334102 Numbers n for which A329697(n) == 2.

Original entry on oeis.org

7, 9, 11, 13, 14, 15, 18, 22, 25, 26, 28, 30, 36, 41, 44, 50, 51, 52, 56, 60, 72, 82, 85, 88, 97, 100, 102, 104, 112, 120, 137, 144, 164, 170, 176, 193, 194, 200, 204, 208, 224, 240, 274, 288, 289, 328, 340, 352, 386, 388, 400, 408, 416, 448, 480, 548, 576, 578, 641, 656, 680, 704, 769, 771, 772, 776, 800, 816, 832, 896, 960, 1096

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Numbers n for which A171462(n) = n-A052126(n) is in A334101.
Numbers k such that A000265(k) is either in A333788 or in A334092.
Each term is either of the form A334092(n)*2^k, for some n >= 1, and k >= 0, or a product of two terms of A334101, whether distinct or not.
Binary weight (A000120) of these terms is always either 2, 3 or 4. It is 2 for those terms that are of the form 9*2^k, 4 for the terms of the form p*q*2^k, where p and q are two distinct Fermat primes (A019434), and 3 for the both terms of the form A334092(n)*2^k, and for the terms of the form (p^2)*(2^k), where p is a Fermat prime > 3.

Row 2 of A334100.
Cf. A000120, A000265, A019434, A052126, A171462, A209229, A329697.
Cf. A333788 (a subsequence), A334092 (primes present), A334093 (primes that are 1 + some term in this sequence).
Squares of A334101 form a subsequence of this sequence. Squares of these numbers can be found (as a subset) in A334104, and the cubes in A334106.

A000265(n) = (n>>valuation(n,2));
isA019434(n) = ((n>2)&&isprime(n)&&!bitand(n-2,n-1)); \\ Charfun for A019434, Fermat primes.
isA334102(n) = { n = A000265(n); if(isprime(n), isA019434(A000265(n-1)), if(bigomega(n)!=2,0,factorback(apply(isA019434,factor(n)[,1])))); };

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334102(n) = (2==A329697(n));

A144482 Semiprimes that are a product of Mersenne primes.

Original entry on oeis.org

9, 21, 49, 93, 217, 381, 889, 961, 3937, 16129, 24573, 57337, 253921, 393213, 917497, 1040257, 1572861, 3670009, 4063201, 16252897, 16646017, 66584449, 67092481, 1073602561, 4294434817, 6442450941, 15032385529, 17179607041

Offset: 1

G. L. Honaker, Jr., Oct 12 2008

nonn

As the product of any two primes is semiprime by definition, this is also the list of composite numbers n=x*y where both x and y are Mersenne primes. - Christian N. K. Anderson, Mar 25 2013

Christian N. K. Anderson, Table of n, a(n) for n = 1..75

Cf. A000668, A001358, A144856, A333788.

Subsequence of A335882.

Take[Times@@@Tuples[2^# -1&/@MersennePrimeExponent[Range[10]],2]//Union,30] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 30 2020 *)

isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA144482(n) = ((2==bigomega(n))&&isA000668(vecmin(factor(n)[,1]))&&isA000668(vecmax(factor(n)[,1]))); \\ Antti Karttunen, Jun 28 2020

A335912 Numbers k for which A335885(k) = 2.

Original entry on oeis.org

9, 11, 13, 15, 18, 19, 21, 22, 23, 25, 26, 29, 30, 35, 36, 38, 41, 42, 44, 46, 47, 49, 50, 51, 52, 58, 60, 61, 67, 70, 72, 76, 79, 82, 84, 85, 88, 92, 93, 94, 97, 98, 100, 102, 104, 113, 116, 119, 120, 122, 134, 137, 140, 144, 152, 155, 158, 164, 168, 170, 176, 184, 186, 188, 191, 193, 194, 196, 200, 204, 208, 217, 223

Offset: 1

Antti Karttunen, Jun 30 2020

nonn

Numbers n such that when we start from k = n, and apply in some combination the nondeterministic maps k -> k - k/p and k -> k + k/p, (where p can be any of the odd prime factors of k), then for some combination we can reach a power of 2 in exactly two steps (but with no combination allowing 0 or 1 steps).

			For n = 70 = 2*5*7, if we first take p = 7 and apply the map n -> n + (n/p), we obtain 80 = 2^4 * 5. We then take p = 5, and apply the map n -> n - (n/p), to obtain 80-16 = 64 = 2^16. Thus we reached a power of 2 in two steps (and there are no shorter paths), therefore 70 is present in this sequence.
For n = 769, which is a prime, 769 - (769/769) yields 768 = 3 * 256. For 768 we can then apply either map to obtain a power of 2, as 768 - (768/3) = 512 = 2^9 and 768 + (768/3) = 1024 = 2^10. On the other hand, 769 + (769/769) = 770 and A335885(770) = 4, so that route would not lead to any shorter paths, therefore 769 is a term of this sequence.

Row 2 of A335910.
Cf. A000265, A335911, A335885.
Subsequences of semiprimes (union gives all odd semiprimes present): A144482, A333788, A336115.
Cf. also A334092, A335874, A336116, A336117 and A334102, A335882, A336122.

A335885(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); };
isA335912(n) = (2==A335885(n));

A336115 Semiprimes that are product of a Fermat prime and a Mersenne prime.

Original entry on oeis.org

9, 15, 21, 35, 51, 93, 119, 155, 381, 527, 635, 771, 1799, 2159, 7967, 24573, 32639, 40955, 139247, 196611, 393213, 458759, 655355, 1572861, 2031647, 2105087, 2228207, 2621435, 8323199, 8912879, 33685247, 134741759, 536813567, 6442450941, 8590000127, 10737418235

Offset: 1

Antti Karttunen, Jul 09 2020

nonn

As 3 is both a Fermat prime and a Mersenne prime, A019434(1) * A000668(1) = 9 is also a term. It is the only square in this sequence.

Cf. A000265, A000668, A019434, A144482, A333788, A335885.

Subsequence of A335912.

isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA019434(n) = ((n>2)&&isprime(n)&&!bitand(n-2,n-1));
isA336115(n) = if(9==n,1, if((2!=omega(n))||(2!=bigomega(n)),0,my(ps=factor(n)[,1]); (isA019434(ps[1])&&isA000668(ps[2]))||(isA019434(ps[2])&&isA000668(ps[1])))); \\ Antti Karttunen, Jul 11 2020

A335885(a(n)) = 2.

Missing terms and more terms added by Jinyuan Wang, Jul 11 2020

A344780 Semiprimes that are product of two distinct Honaker primes.

Original entry on oeis.org

34453, 59867, 120191, 136109, 137419, 142921, 170431, 178291, 187723, 205801, 250603, 253223, 273257, 275887, 280471, 286933, 290951, 297763, 319771, 339421, 342163, 348853, 354617, 356189, 357499, 357943, 367193, 376879, 401777, 410947, 413173, 422999, 449723

Offset: 1

K. D. Bajpai, May 28 2021

Subsequence of A006881.

a(1) = 34453 is the only number <= 5*10^6 that is a triangular number.

			34453 = 131*263 which are distinct Honaker primes.
120191 = 263*457 which are distinct Honaker primes.

Cf. A006881, A033548, A144482, A144856, A333788.

isA006881 := proc(n)
    if numtheory[bigomega](n) =2 and A001221(n) = 2 then
        true ;
    else
        false ;
    end if;
end proc:
isA344780 := proc(n)
    if isA006881(n) then
        for p in ifactors(n)[2] do
            if not isA033548(op(1,p)) then
                return false;
            end if;
        end do:
        true ;
    else
        false;
    end if;
end proc:
for n from 1  do
    if isA344780(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jul 07 2021

fHQ[n_] := Plus @@ IntegerDigits@n == Plus @@ IntegerDigits@PrimePi@n;
lst = {}; Do[If[Plus @@ Last /@ FactorInteger[n] == 2, a = Length[First /@ FactorInteger[n]]; If[a == 2, b = First /@ FactorInteger[n]; c = b[[1]]; d = b[[2]]; If[fHQ[c] && fHQ[d], AppendTo[lst, {n,c,d}]]]], {n, 2000000}]; lst

cp's OEIS Frontend

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

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A334102 Numbers n for which A329697(n) == 2.

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A144482 Semiprimes that are a product of Mersenne primes.

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A335912 Numbers k for which A335885(k) = 2.

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A336115 Semiprimes that are product of a Fermat prime and a Mersenne prime.

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A344780 Semiprimes that are product of two distinct Honaker primes.

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