A050922 -id:A050922 - cp's OEIS frontend

A343767 a(n) is the index of A023394(n) in flattened array A050922.

0, 1, 2, 3, 5, 4, 25, 7, 20, 21, 13, 6

Offset: 1

Lorenzo Sauras Altuzarra, Apr 28 2021

a(14) = 26, a(15) = 16, a(16) = 27, a(20) = 17, a(22) = 28.

Permutation of the natural numbers.

			A023394(1) = 3 = A050922(0), so a(1) = 0.
A023394(2) = 5 = A050922(1), so a(2) = 1.

Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
Index entries for sequences that are related to primes dividing Fermat numbers
Index entries for sequences that are permutations of the natural numbers

Cf. A023394, A050922.

A023394(n) = A050922(a(n)).

A067387 Duplicate of A050922.

Original entry on oeis.org

3, 5, 17, 257, 65537, 641, 6700417, 274177, 67280421310721, 59649589127497217

Offset: 1

dead

A000215 Fermat numbers: a(n) = 2^(2^n) + 1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937

Offset: 0

N. J. A. Sloane

It is conjectured that just the first 5 numbers in this sequence are primes.
An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004
For n>0, Fermat numbers F(n) have digital roots 5 or 8 depending on whether n is even or odd (Koshy). - Lekraj Beedassy, Mar 17 2005
This is the special case k=2 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058. - Seppo Mustonen, Sep 04 2005
For n>1 final two digits of a(n) are periodically repeated with period 4: {17, 57, 37, 97}. - Alexander Adamchuk, Apr 07 2007
For 1 < k <= 2^n, a(A007814(k-1)) divides a(n) + 2^k. More generally, for any number k, let r = k mod 2^n and suppose r != 1, then a(A007814(r-1)) divides a(n) + 2^k. - T. D. Noe, Jul 12 2007
From Daniel Forgues, Jun 20 2011: (Start)
The Fermat numbers F_n are F_n(a,b) = a^(2^n) + b^(2^n) with a = 2 and b = 1.
For n >= 2, all factors of F_n = 2^(2^n) + 1 are of the form k*(2^(n+2)) + 1 (k >= 1).
The products of distinct Fermat numbers (in their binary representation, see A080176) give rows of Sierpiński's triangle (A006943). (End)
Let F(n) be a Fermat number. For n > 2, F(n) is prime if and only if 5^((F(n)-1)/4) == sqrt(F(n)-1) (mod F(n)). - Arkadiusz Wesolowski, Jul 16 2011
Conjecture: let the smallest prime factor of Fermat number F(n) be P(F(n)). If F(n) is composite, then P(F(n)) < 3*2^(2^n/2 - n - 2). - Arkadiusz Wesolowski, Aug 10 2012
The Fermat primes are not Brazilian numbers, so they belong to A220627, but the Fermat composites are Brazilian numbers so they belong to A220571. For a proof, see Proposition 3 page 36 on "Les nombres brésiliens" in Links. - Bernard Schott, Dec 29 2012
It appears that this sequence is generated by starting with a(0)=3 and following the rule "Write in binary and read in base 4".  For an example of "Write in binary and read in ternary", see A014118. - John W. Layman, Jul 30 2013
Conjecture: the numbers > 5 in this sequence, i.e., 2^2^k + 1 for k>1, are exactly the numbers n such that (n-1)^4-1 divides 2^(n-1)-1. - M. F. Hasler, Jul 24 2015

			a(0) = 1*2^1 + 1 = 3 = 1*(2*1) + 1.
a(1) = 1*2^2 + 1 = 5 = 1*(2*2) + 1.
a(2) = 1*2^4 + 1 = 17 = 2*(2*4) + 1.
a(3) = 1*2^8 + 1 = 257 = 16*(2*8) + 1.
a(4) = 1*2^16 + 1 = 65537 = 2048*(2*16) + 1.
a(5) = 1*2^32 + 1 = 4294967297 = 641*6700417 = (10*(2*32) + 1)*(104694*(2*32) + 1).
a(6) = 1*2^64 + 1 = 18446744073709551617 = 274177*67280421310721 = (2142*(2*64) + 1)*(525628291490*(2*64) + 1).

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 7.
P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 87.
James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.
R. K. Guy, Unsolved Problems in Number Theory, A3.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 14.
E. Hille, Analytic Function Theory, Vol. I, Chelsea, N.Y., see p. 64.
T. Koshy, "The Digital Root Of A Fermat Number", Journal of Recreational Mathematics Vol. 32 No. 2 2002-3 Baywood NY.
M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001.
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, p. 36.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 18, 59.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 202.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 6-7, 70-75.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 148-149.

N. J. A. Sloane, Table of n, a(n) for n = 0..11
Richard Bellman, A note on relatively prime sequences, Bull. Amer. Math. Soc. 53 (1947), 778-779.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Chris K. Caldwell, The Prime Glossary, Fermat number.
Chris K. Caldwell, The prime pages All prime-square Mersenne divisors are Wieferich (2021).
Shane Chern, Fermat Numbers in Multinomial Coefficients, J. Int. Seq. 17 (2014) # 14.3.2.
Leonhard Euler, Observations on a theorem of Fermat and others on looking at prime numbers, arXiv:math/0501118 [math.HO], 2005-2008.
Leonhard Euler, Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus.
Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 12.
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
Jiří Klaška, Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.
T.-W. Leung, A Brief Introduction to Fermat Numbers
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From _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.
Michael A. Morrison and John Brillhart, The factorization of F_7, Bull. Amer. Math. Soc. 77 (1971), 264.
Robert Munafo, Fermat Numbers
Robert Munafo, Notes on Fermat numbers
Seppo Mustonen, On integer sequences with mutual k-residues.
Seppo Mustonen, On integer sequences with mutual k-residues. [Local copy]
OEIS Wiki, Fermat numbers.
OEIS Wiki, Sierpinski's triangle.
G. A. Paxson, The compositeness of the thirteenth Fermat number, Math. Comp. 15 (76) (1961) 420-420.
Carl Pomerance, A tale of two sieves, Notices Amer. Math. Soc., 43 (1996), 1473-1485.
P. Sanchez, PlanetMath.org, Fermat Numbers
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.
G. Villemin's Almanach of Numbers, Nombres de Fermat.
Le Roy J. Warren, Henry G. Bray, On the square-freeness of Fermat and Mersenne Numbers, Pac. J. Math. 22 (3) (1967) 563.
Eric Weisstein's World of Mathematics, Fermat Number.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
Wikipedia, Fermat number.
Wolfram Research, Fermat numbers are pairwise coprime.

a(n) = A001146(n) + 1 = A051179(n) + 2.
Cf. A019434, A050922, A051158, A051179, A063486, A073617, A085866.
See A004249 for a similar sequence.
Cf. A080176 for binary representation of Fermat numbers.
Cf. A220627, A220570, A220571, A125134, A249119.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.

a000215 = (+ 1) . (2 ^) . (2 ^)  -- Reinhard Zumkeller, Feb 13 2015

Maple
```
A000215 := n->2^(2^n)+1;
```

Table[2^(2^n) + 1, {n, 0, 8}] (* Alonso del Arte, Jun 07 2011 *)

A000215(n):=2^(2^n)+1$ makelist(A000215(n),n,0,10); /* Martin Ettl, Dec 10 2012 */

PARI
```
a(n)=if(n<1,3*(n==0),(a(n-1)-1)^2+1)
```

def a(n): return 2**(2**n) + 1
print([a(n) for n in range(9)]) # Michael S. Branicky, Apr 19 2021

a(0) = 3; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
a(n) = a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get the empty product, i.e., 1, plus 2, giving 3 = a(0). - Benoit Cloitre, Sep 15 2002 [edited by Daniel Forgues, Jun 20 2011]
The above formula implies that the Fermat numbers (being all odd) are coprime.
Conjecture: F is a Fermat prime if and only if phi(F-2) = (F-1)/2. - Benoit Cloitre, Sep 15 2002
A000120(a(n)) = 2. - Reinhard Zumkeller, Aug 07 2010
If a(n) is composite, then a(n) = A242619(n)^2 + A242620(n)^2 = A257916(n)^2 - A257917(n)^2. - Arkadiusz Wesolowski, May 13 2015
Sum_{n>=0} 1/a(n) = A051158. - Amiram Eldar, Oct 27 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = A249119.
Product_{n>=0} (1 - 1/a(n)) = 1/2. (End)
a(n) = 2*A077585(n) + 3. - César Aguilera, Jul 26 2023
a(n) = 2*2^A000225(n) + 1. - César Aguilera, Jul 11 2024

A002586 Smallest prime factor of 2^n + 1.

Original entry on oeis.org

3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 65537, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 274177, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 65537, 3, 5, 3, 17, 3, 5

Offset: 1

N. J. A. Sloane

Conjecture: a(8+48*k) = 257 and a(40+48*k) = 257, where k is a nonnegative integer. - Thomas König, Feb 15 2017
Conjecture is true: 257 divides 2^(8+48*k)+1 and 2^(40+48*k)+1 but no prime < 257 ever does. Similarly, a(24+48*k) = 97. - Robert Israel, Feb 17 2017
From Robert Israel, Feb 17 2017: (Start)
If a(n) = p, there is some m such that a(n+m*j*n) = p for all j.
In particular, every member of the sequence occurs infinitely often.
a(k*n) <= a(n) for any odd k. (End)

			a(2^k) = 3, 5, 17, 257, 65537 is the k-th Fermat prime 2^(2^k) + 1 = A019434(k) for k = 0, 1, 2, 3, 4. - _Jonathan Sondow_, Nov 28 2012

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres, Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of n, a(n) for n = 1..3967 (first 500 terms from T. D. Noe)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Thomas König, C program using gmp for testing the conjectures that a(8+k*48) = 257 and a(40+k*48) = 257
E. Lucas, Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240, 289-321. See pages 239 and 240.
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000215, A001269, A002587, A019434, A050922, A093179.

f[n_] := FactorInteger[2^n + 1][[1, 1]]; Array[f, 100] (* Robert G. Wilson v, Nov 28 2012 *)
FactorInteger[#][[1,1]]&/@(2^Range[90]+1) (* Harvey P. Dale, Jul 25 2024 *)

a(n) = my(m=n%8); if(m, [3, 5, 3, 17, 3, 5, 3][m], factor(2^n+1)[1,1]); \\ Ruud H.G. van Tol, Feb 16 2024

from sympy import primefactors
smallest_primef = []
for n in range(1,87):
    y = (2 ** n) + 1
    smallest_primef.append(min(primefactors(y)))
print(smallest_primef) # Adrienne Leonardo, Dec 29 2024

a(n) = 3, 5, 3, 17, 3, 5, 3 for n == 1, 2, 3, 4, 5, 6, 7 (mod 8). (Proof. Let n = k*odd with k = 1, 2, or 4. As 2^k = 2, 4, 16 == -1 (mod 3, 5, 17), we get 2^n + 1 = 2^(k*odd) + 1 = (2^k)^odd + 1 == (-1)^odd + 1 == 0 (mod 3, 5, 17). Finally, 2^n + 1 !== 0 (mod p) for prime p < 3, 5, 17, respectively.) - Jonathan Sondow, Nov 28 2012

More terms from James Sellers, Jul 06 2000

Definition corrected by Jonathan Sondow, Nov 27 2012

A093179 Smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 641, 274177, 59649589127497217, 1238926361552897, 2424833, 45592577, 319489, 114689, 2710954639361, 116928085873074369829035993834596371340386703423373313, 1214251009, 825753601, 31065037602817, 13631489, 70525124609

Offset: 0

Eric W. Weisstein, Mar 27 2004

a(14) might need to be corrected if F(14) turns out to have a smaller factor than 116928085873074369829035993834596371340386703423373313. F(20) is composite, but no explicit factor is known. - Jeppe Stig Nielsen, Feb 11 2010

			F(0) = 2^(2^0) + 1 = 3, prime.
F(5) = 2^(2^5) + 1 = 4294967297 = 641*6700417.
So 3 as the 0th entry and 641 is the 5th term.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 73.

Wilfrid Keller, Prime factors k·2^n + 1 of Fermat numbers F_m and complete factoring status
Ivars Peterson, Cracking Fermat Numbers
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000051, A000215, A020639, A070592, A007117.

Leading entries in triangle A050922.

Table[With[{k = 2^n}, FactorInteger[2^k + 1]][[1, 1]], {n, 0, 15, 1}] (* Vincenzo Librandi, Jul 23 2013 *)

g(n)=for(x=9,n,y=Vec(ifactor(2^(2^x)+1));print1(y[1]",")) \\ Cino Hilliard, Jul 04 2007

a(n) = A007117(n)*2^(n+2) + 1 for n >= 2. - Jianing Song, Mar 02 2021

a(n) = A020639(A000215(n)). - Alois P. Heinz, Oct 25 2024

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar
a(14)-a(15) added by Jeppe Stig Nielsen, Feb 11 2010
a(16)-a(19) added based on terms of A007117 by Jianing Song, Mar 02 2021

A070592 Largest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 6700417, 67280421310721, 5704689200685129054721, 93461639715357977769163558199606896584051237541638188580280321

Offset: 0

Benoit Cloitre, May 12 2002

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 72.

Eric M. Schmidt, Table of n, a(n) for n = 0..11
C. L. Stewart, On Divisors of Fermat, Fibonacci, Lucas, and Lehmer Numbers, Proceedings of the London Mathematical Society, Vol. s3-35, No. 3 (1977), pp. 425-447. See p. 430.
Eric Weisstein's World of Mathematics, Fermat Number.

Cf. A000215, A006530, A050922, A093179.

a(n) = vecmax(factor(2^(2^n) + 1)[,1]); \\ Michel Marcus, Jul 05 2017

From Amiram Eldar, Oct 25 2024: (Start)
a(n) = A006530(A000215(n)).
a(n) > c * n * 2^n for n >= 1, where c is a positive absolute constant (Stewart, 1977). (End)

Offset changed by Arkadiusz Wesolowski, Jul 09 2011

A307843 Divisors of Fermat numbers.

Original entry on oeis.org

1, 3, 5, 17, 257, 641, 65537, 114689, 274177, 319489, 974849, 2424833, 6700417, 13631489, 26017793, 45592577, 63766529, 167772161, 825753601, 1214251009, 4294967297, 6487031809, 70525124609, 190274191361, 311453532161, 646730219521, 2710954639361

Offset: 1

Jeppe Stig Nielsen, Jul 24 2019

nonn

Has both A000215 and A023394 as subsequences. Outside these are 1 and the composite proper divisors of Fermat numbers, namely 311453532161, 2983954661377, 7313319444481, ...

Odd m = (p_1)^(e_1)*(p_2)^(e_2)*...*(p_r)^(e_r) is a term if and only if the multiplicative order of 2 modulo (p_i)^(e_i) is the same power of 2 for 1 <= i <= r. - Jianing Song, May 19 2024

			311453532161 is included because it divides 2^(2^11) + 1. It is not included in A023394 because it is composite.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..58

Cf. A000215, A023394, A050922, A094358.

isA307843(n) = if(n==1, return(1)); if(n%2, my(f = factor(n), d = znorder(Mod(2,f[1,1]^f[1,2]))); if(!isprimepower(2*d), return(0)); for(i=2, #f~, if(znorder(Mod(2,f[i,1]^f[i,2])) != d, return(0))); 1, 0) \\ Jianing Song, May 19 2024. Inefficient to print the sequence as terms are sparse

A155877 Sums of three Fermat numbers.

Original entry on oeis.org

9, 11, 13, 15, 23, 25, 27, 37, 39, 51, 263, 265, 267, 277, 279, 291, 517, 519, 531, 771, 65543, 65545, 65547, 65557, 65559, 65571, 65797, 65799, 65811, 66051, 131077, 131079, 131091, 131331, 196611, 4294967303, 4294967305, 4294967307

Offset: 1

Jonathan Vos Post, Jan 29 2009

nonn

			a(1) = 3 + 3 + 3 = 9.
a(2) = 3 + 3 + 5 = 11.
a(3) = 3 + 5 + 5 = 13.
a(4) = 5 + 5 + 5 = 15.
a(5) = 3 + 3 + 17 = 23.
a(6) = 3 + 5 + 17 = 25.
a(7) = 5 + 5 + 17 = 27.
a(8) = 3 + 17 + 17 = 37.
a(9) = 5 + 17 + 17 = 39.
a(10) = 17 + 17 + 17 = 51.
a(11) = 3 + 3 + 257 = 263.

Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A000215, A019434, A050922, A051179, A059919, A063486, A073617, A078303, A078304, A085866.

{(2^(2^a) + 1) + (2^(2^b) + 1) + (2^(2^c) + 1)} = {A000215(a) + A000215(b) + A000215(c)}.

More terms from R. J. Mathar, Feb 06 2009

A176689 Prime factors of 2^128 - 1.

Original entry on oeis.org

3, 5, 17, 257, 641, 65537, 274177, 6700417, 67280421310721

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 23 2010

Cf. A000215, A002590, A019434, A023394, A050922, A070592.

Mathematica
```
First/@FactorInteger[2^128-1]
```

factor(2^128 - 1)[,1] \\ Charles R Greathouse IV, Sep 06 2016

Edited by T. D. Noe, May 06 2010

Typo in definition corrected by Arkadiusz Wesolowski, Feb 17 2011

A281577 Irregular triangle read by rows: T(n, k) = A281576(n) modulo p^2, where p is the k-th prime factor of A281576(n) with p < sqrt(A281576(n)).

Original entry on oeis.org

28204, 17161560961, 2451293172821355028751076998879853, 1409441895293467096954080352837, 1385195550582, 17782786311867894562037823351528977990025091057921642664123352687840735480821116989430796689072791

Offset: 1

Felix Fröhlich, Jan 24 2017

The question whether T(n, k) = 0 for any values of n and k is an open problem (see Ribenboim p. 64, open problem (3)).

			Triangle T(n, k) starts
                                  28204
                            17161560961
     2451293172821355028751076998879853
        1409441895293467096954080352837
                 1385195550582, T(5, 2)
Note: T(5, 2) is not displayed here due to its size. The term can be seen in the Data section.

P. Ribenboim, The Little Book of Bigger Primes, Springer Verlag, 1991.

Cf. A050922, A281576.

a152155(n) = centerlift(Mod(3, 2^(2^n)+1)^(2^(2^n-1)))
row(n) = my(i=0, k=1); while(1, if(a152155(k)!=-1, i++); if(i==n, forprime(p=1, sqrtint(2^(2^k)+1), if(Mod(2, p)^(2^k)==-1, print1(lift(Mod(2, p^2)^(2^k))+1, ", ")))); k++)
trianglerows(n) = for(k=1, n, row(k); print(""))

cp's OEIS Frontend

A343767 a(n) is the index of A023394(n) in flattened array A050922.

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A067387 Duplicate of A050922.

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A000215 Fermat numbers: a(n) = 2^(2^n) + 1.

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A002586 Smallest prime factor of 2^n + 1.

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A093179 Smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

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A070592 Largest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

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A307843 Divisors of Fermat numbers.

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