A093179 -id:A093179 - cp's OEIS frontend

A259534 a(n) = -1 + 2 * product_{i=0..n} A093179(i), where A093179(i) is the smallest prime factor of 2^(2^i) + 1.

5, 29, 509, 131069, 8589934589, 5506148072189, 1509659159988837629, 90050548615896750734368618889875709, 111565998552535226317138856424609779410946920431869, 270528914968139650436266764640655805238384653911572627709

Offset: 0

Arkadiusz Wesolowski, Jul 02 2015

nonn

For any k >= 1, numbers of the form (k*a(n) + k - 1)*2^m - 1 are composite for all m < 2^(n+1).

Many terms are in common with A006285 (de Polignac numbers).

Cf. A006285, A093179.

a(n) = - 1 + 2 * prod(k=0, n, factor(2^(2^k)+1)[1,1]); \\ Michel Marcus, Jul 04 2015

A360469 Only k >= 0 such that, for every odd r > 0, A093179(n) divides the generalized Fermat number (A007117(n)^r)^(2^k) + 1.

Original entry on oeis.org

3, 3, 5, 3, 7, 7, 9, 8, 11, 11, 13, 10, 15, 15, 17, 16, 19, 19, 21, 19, 23, 23, 25, 24, 27, 27, 29, 25, 31, 31, 33, 32, 35, 35, 37, 35, 39, 39, 41, 40, 43, 43, 45, 42, 47, 47, 49, 48, 51, 51, 53, 51, 55, 55, 57, 56, 59, 59, 61, 56, 63, 63, 65, 64, 67, 67, 69, 67, 71, 71, 73, 72, 75, 75, 77, 74, 79

Offset: 3

Lorenzo Sauras Altuzarra, Feb 27 2023

			A093179(5) = 641, A007117(5) = 5 and the only k >= 0 such that, for every odd r > 0, 641 divides the generalized Fermat number (5^r)^(2^k) + 1 is 5; so a(5) = 5.

Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).

Cf. A000215 (Fermat numbers), A007117, A007814 (dyadic valuation), A093179, A307843 (divisors of Fermat numbers).

a:=n->n-padic:-ordp(n+2,2):
seq(a(n), n=3..79);

a(n) = n - valuation(n+2, 2);
vector(77,n,a(n+2)) \\ Joerg Arndt, Mar 03 2023

a(n) = n - A007814(n + 2) (due to Jinyuan Wang).

A092559 Numbers k such that 2^k + 1 is a semiprime.

Original entry on oeis.org

3, 5, 6, 7, 11, 12, 13, 17, 19, 20, 23, 28, 31, 32, 40, 43, 61, 64, 79, 92, 101, 104, 127, 128, 148, 167, 191, 199, 256, 313, 347, 356, 596, 692, 701, 1004, 1228, 1268, 1709, 2617, 3539, 3824, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239

Offset: 1

Zak Seidov, Feb 27 2004

nonn

Thanks to the recently found factor of F_14 (see A093179), we know that 16384 is not in the sequence. First unknown: 16768. - Don Reble, Mar 28 2010
The big prime factors for "5807" and all smaller entries have been proved prime; the rest (as far as I know) are probable primes. - Don Reble, Mar 28 2010
From Giuseppe Coppoletta, May 09 2017: (Start)
As 3 divides 2^a(n) + 1 for any odd a(n), all odd terms are prime and they are exactly the Wagstaff numbers (A000978) or also the prime Jacobsthal indices (A107036).
All terms from a(51) onwards refer to probabilistic primality tests for 2^a(n) + 1 (see Caldwell's link for the list of the largest certified Wagstaff primes).
For the close relationship between this sequence and the Fermat numbers, see comments in A073936. The only difference is that here a term can be the square of a prime p, and by the Mihăilescu Theorem (also known as Catalan's conjecture, see link) that implies p = a(n) = 3. So, excluding a(1) = 3, they must coincide.
As for A073936, after a(57), the values 267017, 269987, 374321, 986191, 4031399 and 4101572 are also terms, but there still remains the remote possibility of some gaps in between. In addition, 13347311 and 13372531 are also terms, but are possibly much further along in the numbering (see comments in A000978).
(End).
The powers of 2 in this sequence (that correspond to semiprime Fermat numbers) are k = 2^m with m = 5, 6, 7, 8, and no more below 20. - Amiram Eldar, Jun 18 2022

			11 is a term because 2^11 + 1 = 3 * 683.
3 is a term because 2^3 + 1 = 3^2.
10 is not a term because 2^10 + 1 = 5^2 * 41.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..57
C. Caldwell's The Top Twenty Wagstaff primes.
S. S. Wagstaff, Jr., The Cunningham Project.
Eric W. Weisstein, MathWorld: Catalan's Conjecture.

Cf. A085724, A092558, A092561, A092562, A000978, A107036, A066263.

Cf. A073936. - R. J. Mathar, Sep 08 2008

Select[Range@ 200, PrimeOmega[2^# + 1] == 2 &] (* Michael De Vlieger, May 09 2017 *)

isok(n) = bigomega(2^n+1) == 2; \\ Michel Marcus, Oct 05 2013

More terms from Cunningham project, Mar 23 2004
More terms from Don Reble, Mar 28 2010
a(49)-a(52) from Giuseppe Coppoletta, May 08 2017

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

A050922 Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 641, 6700417, 274177, 67280421310721, 59649589127497217, 5704689200685129054721, 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321

Offset: 0

N. J. A. Sloane, Dec 30 1999

Alternatively, list of prime factors of terms of A001317 in order of their first appearance. - Labos Elemer, Jan 21 2002
From T. D. Noe, Jan 29 2009: (Start)
That these two definitions give the same sequence follows from the fact (stated as a formula in A001317) that A001317(n) is the product of Fermat numbers F(i) according to which bits of n are set.
For instance, for n=41, the binary representation of n is 101001, which has bits 0, 3 and 5 set. A001317(n) = 3311419785987 = 3*257*4294967297 = F(0)*F(3)*F(5).
This factorization also explains why the "first 31 numbers give odd-sided constructible polygons". I think Hewgill first noticed this factorization. (End)

			Triangle begins:
  3;
  5;
  17;
  257;
  65537;
  641,               6700417;
  274177,            67280421310721;
  59649589127497217, 5704689200685129054721;
  1238926361552897,  93461639715357977769163558199606896584051237541638188580280321;
  ...
A001317(127) = 3*5*17*257*65537.641*6700417*274177*6728042130721, A001317(128) = 59649589127497217*5704689200685129054721. See also A050922. Compare with A053576, where 2 and A000215 appear as prime factors. - _Labos Elemer_, Jan 21 2002

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..29
J. Bernheiden, Fermat Numbers (Text in German)
R. P. Brent, Factorization of the tenth Fermat number
R. P. Brent, Factorization of the eleventh Fermat number
R. P. Brent, Succinct proofs of primality for the factors of some Fermat numbers
R. P. Brent & J. M. Pollard, Factorization of the eighth Fermat number
R. P. Brent et al., Three new factors of Fermat numbers
C. K. Caldwell, The Prime Glossary, Fermat divisor
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
R. Mestrovic, 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, 2012. - From _N. J. A. Sloane_, Jun 13 2012
R. Munafo, Notes on Fermat numbers
Mercedes Orús-Lacort, Fermat numbers are not prime numbers for n >= 5, (2020).
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A019434, A023394, A093179.

Cf. A001317, A001316, A003401, A045544, A053576.

Flatten[Transpose[FactorInteger[#]][[1]]&/@Table[2^(2^n)+1,{n,0,8}]] (* Harvey P. Dale, May 18 2012 *)

for(n=0, 1e3, f=factor(2^(2^n)+1)[, 1]; for(i=1, #f, print1(f[i], ", "))) \\ Felix Fröhlich, Aug 16 2014

More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000.
Edited by N. J. A. Sloane, Jan 31 2009 at the suggestion of T. D. Noe
Link to Munafo webpage fixed by Robert Munafo, Dec 09 2009

A007117 a(0) = a(1) = 0; for n >= 2, a(n)*2^(n+2) + 1 is the smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

Original entry on oeis.org

0, 0, 1, 8, 1024, 5, 1071, 116503103764643, 1209889024954, 1184, 11131, 39, 7, 82731770, 1784180997819127957596374417642156545110881094717, 9264, 3150, 59251857, 13, 33629

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

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

			From _Jianing Song_, Mar 02 2021: (Start)
F(2) = 2^(2^2) + 1 = 1*2^4 + 1;
F(3) = 2^(2^3) + 1 = 5*2^5 + 1;
F(4) = 2^(2^4) + 1 = 1024*2^6 + 1;
F(5) = 2^(2^5) + 1 = (5*2^7 + 1) * (52347*2^7 + 1);
F(6) = 2^(2^6) + 1 = (1071*2^8 + 1) * (262814145745*2^8 + 1). (End)

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 71.
H. Riesel, ``Prime numbers and computer methods for factorization,'' Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 377.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
R. G. Wilson, V, Letter to N. J. A. Sloane, Aug. 1993

Cf. A093179.

a(n) = if(n<2, 0, my(lim=2^(2^n-(n+2))); for(k=1, lim, my(p=k*2^(n+2)+1); if(Mod(2,p)^(2^n)==-1, return(k)))) \\ Jianing Song, Mar 02 2021

a(n) = (A093179(n) - 1)/2^(n+2) for n >= 2. - Jianing Song, Mar 02 2021

a(14)-a(19) added by Max Alekseyev, May 04 2010

A185121 Smallest prime factor of 10^(2^n) + 1.

Original entry on oeis.org

11, 101, 73, 17, 353, 19841, 1265011073, 257, 10753, 1514497, 1856104284667693057, 106907803649, 458924033, 3635898263938497962802538435084289

Offset: 0

Sergio Pimentel, Jan 22 2012

10^k+1 can only be prime if k is a power of 2. So far the only known primes of this form are a(0) = 11 and a(1) = 101. [Edited by M. F. Hasler, Aug 03 2019]
a(n) >= 2^(n+1)+1; we have a(n) = 2^(n+1)+1 for n=3, n=7, and n=15.
a(14) > 10^16. - Max Alekseyev, Jun 28 2013
From the Keller link a(15)-a(20) = 65537, 8257537, 175636481, 639631361, 70254593, 167772161. - Ray Chandler, Dec 27 2013

			For n=2, a(2)=73 since 10^(2^2) + 1 = 10001 = 73 * 137.

FactorDB, Factorizations of 10^(2^n)+1
Makoto Kamada, Factorizations of 100...001
Wilfrid Keller, Prime factors of generalized Fermat numbers Fm(10) and complete factoring status

Cf. A038371, A000533, A000215, A093179, A102050

Essentially the same as A102050. - Sean A. Irvine, Feb 17 2013

Table[With[{k = 2^n}, FactorInteger[10^k + 1]][[1, 1]], {n, 0, 13, 1}] (* Vincenzo Librandi, Jul 23 2013 *)

a(n) = factor(10^(2^n)+1)[1, 1] \\ Michel Marcus, May 30 2013

a(n) = A038371(2^n). - M. F. Hasler, Jul 30 2019

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

A164307 Primes in A081175.

Original entry on oeis.org

3, 5, 17, 257, 65537

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

The 6th term is too large to include in the data section (see Example section or the b-file).
Primes of the form sum_{j=1..u} j^x for some x>0, u>1. (Since the case of x=1 leads to the triangular numbers with no additional primes, this is equivalent to the definition.)
Primes in A000330 (x=2), or in A000537 (x=3), or in A000538 (x=4), or in A000539 (x=5) etc. See A164312 for the corresponding x values.

			a(1) = 1^1 + 2^1 = 3.
a(2) = 1^2 + 2^2 = 5.
a(3) = 1^4 + 2^4 = 17.
a(4) = 1^8 + 2^8 = 257.
a(5) = 1^16 + 2^16 = 65537.
a(6) = 1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379.

N. J. A. Sloane, Table of n, a(n) for n = 1..6

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599.

lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,s];Print[Date[],s]],{n, 4!}],{x,7!}];lst

Edited by R. J. Mathar, Aug 22 2009

Corrected by N. J. A. Sloane, Nov 23 2015 at the suggestion of Jaroslav Krizek.

A257916 a(n) is the largest x that is a member of a pair (x, y) of integers with x - y > 1 such that x^2 - y^2 is equal to the Fermat number 2^(2^n) + 1, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 3350529, 33640210792449, 2852374425137128275969, 46730819857678988884581779099803448292025618771438557470916609

Offset: 0

Arkadiusz Wesolowski, May 12 2015

nonn

2^(2^n) + 1 belongs to A019434 if and only if a(n) = 0.

M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, p. 6.

Wikipedia, Fermat number

Cf. A000215, A257917.

a(n) = {my(fn = 2^(2^n) + 1); if (isprime(fn), return (0)); my(spf = factor(fn)[1,1]); (fn/spf + spf)/2;} \\ Michel Marcus, Jun 07 2015

If F(n) = 2^(2^n) + 1 is composite, then a(n) = (A032742(F(n)) + A093179(n))/2.

cp's OEIS Frontend

A259534 a(n) = -1 + 2 * product_{i=0..n} A093179(i), where A093179(i) is the smallest prime factor of 2^(2^i) + 1.

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A360469 Only k >= 0 such that, for every odd r > 0, A093179(n) divides the generalized Fermat number (A007117(n)^r)^(2^k) + 1.

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A092559 Numbers k such that 2^k + 1 is a semiprime.

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A002586 Smallest prime factor of 2^n + 1.

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A050922 Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1.

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A007117 a(0) = a(1) = 0; for n >= 2, a(n)*2^(n+2) + 1 is the smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

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A185121 Smallest prime factor of 10^(2^n) + 1.

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