A023394 -id:A023394 - cp's OEIS frontend

A094358 Squarefree products of factors of Fermat numbers (A023394).

1, 3, 5, 15, 17, 51, 85, 255, 257, 641, 771, 1285, 1923, 3205, 3855, 4369, 9615, 10897, 13107, 21845, 32691, 54485, 65535, 65537, 114689, 163455, 164737, 196611, 274177, 319489, 327685, 344067, 494211, 573445, 822531, 823685, 958467, 974849, 983055

Offset: 1

Robert Munafo, Apr 26 2004

nonn

641 is the first member not in sequences A001317, A004729, etc.
Conjectured (by Munafo, see link) to be the same as: numbers n such that 2^^n == 1 mod n, where 2^^n is A014221(n).
It is clear from the observations by Max Alekseyev in A023394 and the Chinese remainder theorem that any squarefree product b of divisors of Fermat numbers satisfies 2^(2^b) == 1 (mod b), hence satisfies Munafo's congruence above. The converse is true iff all Fermat numbers are squarefree. However, if nonsquarefree Fermat numbers exist, the criterion that is equivalent with Munafo's property would be "numbers b such that each prime power that divides b also divides some Fermat number". - Jeppe Stig Nielsen, Mar 05 2014
Also numbers b such that b is (squarefree and) a divisor of A051179(m) for some m. Or odd (squarefree) b where the multiplicative order of 2 mod b is a power of two. - Jeppe Stig Nielsen, Mar 07 2014
From Jianing Song, Nov 11 2023: (Start)
Also squarefree numbers k such that there exists i >= 1 such that k divides 2^^i - 1, where 2^^i = 2^2^...^2 (i times) = A014221(i): 2^^i == 1 (mod k) if and only if ord(2,k) divides 2^^(i-1) (ord(a,k) is the multiplicative order of a modulo k), so such i exists if and only if ord(2,k) is a power of 2. For such k, k divides 2^^i - 1 if and only if 2^^(i-2) >= log_2(ord(2,k)).
Note that 2^^(i-1) divides 2^^i implies that 2^^i - 1 divides 2^^(i+1) - 1, so this sequence is also squarefree numbers k such that k divides 2^^i - 1 for all sufficiently large i. (End)

			3 is a term because it is in A023394.
51 is a term because it is 3*17 and 17 is also in A023394.
153 = 3*3*17 is not a term because its factorization includes two 3's.
See the Munafo link for examples of the (conjectured) 2^^n == 1 (mod n) property.

Robert G. Wilson v, T. D. Noe and Ray Chandler, Table of n, a(n) for n = 1..3393 (Original 55 terms from Robert G. Wilson, extended to 1314 terms from T. D. Noe)
Sourangshu Ghosh and Pranjal Jain, On Fermat Numbers and Munafo's Conjecture, (2021).
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 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Robert Munafo, Sequence A094358, 2^^A(N) = 1 mod N

Cf. A023394, A014221, A092188, A001317, A004729.

kmax = 10^6;
A023394 = Select[Prime[Range[kmax]], IntegerQ[Log[2, MultiplicativeOrder[2, #] ] ]&];
Reap[For[k = 1, k <= kmax, k++, ff = FactorInteger[k]; If[k == 1 || AllTrue[ff, MemberQ[A023394, #[[1]]] && #[[2]] == 1 &], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Nov 03 2018 *)

(  isOK1(n) = n%2==1 && hammingweight(znorder(Mod(2,n)))==1  ) ;  (  isOK2(n) = issquarefree(n) && isOK1(n)  )  \\ isOK1 and isOK2 differ only if n contains a squared prime that divides a Fermat number (none are known) \\ Jeppe Stig Nielsen, Apr 02 2014

Edited by T. D. Noe, Feb 02 2009

Example brought in line with name/description by Robert Munafo, May 18 2011

A023395 Only Fermat number divisible by A023394(n) is 2^2^a(n) + 1.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 12, 6, 11, 11, 9, 5, 18, 12, 10, 12, 23, 16, 15, 10, 19, 12, 19, 13, 36, 21, 38, 32, 25, 17, 39, 6, 26, 27, 30, 30, 8, 12, 15, 29, 38, 7, 25, 27, 36, 42, 25, 13, 13, 55

Offset: 1

David W. Wilson

From Jianing Song, Mar 02 2021: (Start)
2^(a(n)+1) is the multiplicative order of 2 modulo A023394(n).
Each k occurs A046052(k) times in this sequence provided that F(k) = 2^2^k + 1 is squarefree (no counterexamples are known). (End)
Alternatively, a(n) is the only k such that A023394(n) divides A000215(k). - Lorenzo Sauras Altuzarra, Feb 01 2023

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
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

Cf. A000215, A023394, A046052.

forprime(p=3,,r=znorder(Mod(2,p));hammingweight(r)==1&&print1(logint(r,2)-1,", ")) \\ Jeppe Stig Nielsen, Mar 04 2018

a(25)-a(41) computed using data from Wilfrid Keller by T. D. Noe, Feb 01 2009
Three more terms by T. D. Noe, Feb 03 2009
Six more terms from Wilfrid Keller by T. D. Noe, Jan 14 2013

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

Original entry on oeis.org

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)).

A344784 Decimal expansion of the sum of the reciprocals of the prime factors of Fermat numbers (A023394).

Original entry on oeis.org

5, 9, 7, 6, 4, 0, 4, 7, 5, 8

Offset: 0

Amiram Eldar, May 28 2021

Golomb (1955) asked if this series is convergent. Křížek et al. (2002) proved its convergence.

The first 10 terms were given by Finch (2018).

			0.5976404758...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, Section 1.37, p. 248.

Steven R. Finch, Fermat Numbers and Elite Primes, preprint, 2013.
Solomon W. Golomb, Sets of primes with intermediate density, Math. Scand., Vol. 3 (1955), pp. 264-274; alternative link.
Michal Křížek, Florian Luca and Lawrence Somer, On the convergence of series of reciprocals of primes related to the Fermat numbers, J. Number Theory, Vol. 97, No. 1 (2002), pp. 95-112.

Cf. A000215, A023394.

Equals Sum_{k>=1} 1/A023394(k).

A372891 Anti-elite primes (A128852) that are not prime factors of Fermat primes (A023394).

Original entry on oeis.org

2, 13, 97, 193, 241, 673, 769, 2689, 5953, 8929, 12289, 40961, 49921, 61681, 101377, 286721, 414721, 417793, 550801, 786433, 1130641, 1376257, 1489153, 1810433, 3602561, 6942721, 7340033, 11304961, 12380161, 15790321, 17047297, 22253377, 39714817, 67411969, 89210881, 93585409, 113246209, 119782433, 152371201, 171048961, 185602561, 377487361, 394783681

Offset: 1

Jianing Song, May 15 2024

nonn

Union of {2} and odd anti-elite primes p such that the multiplicative order of 2 modulo p is not a power of 2.
A128852 is the union of this sequence and prime factors of Fermat numbers >= 17.
Conjecture: All terms >= 97 are congruent to 1 modulo 8. (Note that every factor of Fermat numbers >= 17 is congruent to 1 modulo 8.)

			For n >= 2, we have 2^2^n + 1 == 4 (mod 13) for even n and 2^2^n + 1 == 10 (mod 13) for odd n. As 4 and 10 are both squares modulo 13, and 13 is not a factor of Fermat numbers (the multiplicative order of 2 modulo 13 is 12), 13 is a term.
For n >= 4, we have 2^2^n + 1 == 62 (mod 97) for even n and 2^2^n + 1 == 36 (mod 97) for odd n. As 36 and 62 are both squares modulo 97, and 97 is not a factor of Fermat numbers (the multiplicative order of 2 modulo 97 is 48), 97 is a term.

Cf. A023394, A128852.

isA372891(n) = if(isprime(n) && n > 2, my(d = znorder(Mod(2, n))); if(isprimepower(2*d), return(0)); my(StartPoint = valuation(d, 2), LengthTest = znorder(Mod(2, d >> StartPoint))); for(i = StartPoint, StartPoint + LengthTest - 1, if(!issquare(Mod(2, n)^2^i + 1), return(0))); 1, n == 2)

A245970 Tower of 2's modulo n.

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 2, 0, 7, 6, 9, 4, 3, 2, 1, 0, 1, 16, 5, 16, 16, 20, 6, 16, 11, 16, 7, 16, 25, 16, 2, 0, 31, 18, 16, 16, 9, 24, 16, 16, 18, 16, 4, 20, 16, 6, 17, 16, 23, 36, 1, 16, 28, 34, 31, 16, 43, 54, 48, 16, 22, 2, 16, 0, 16, 64, 17, 52, 52, 16, 3, 16

Offset: 1

Wayne VanWeerthuizen, Aug 08 2014

a(n) = (2^(2^(2^(2^(2^ ... ))))) mod n, provided enough 2's are in the tower so that adding more doesn't affect the value of a(n).

Let b(i) = A014221(i) = (2^(2^(2^(2^(2^ ... ))))), with i 2's. Since gcd(b(i)+1, b(j)+1) = gcd(2^2^b(i-2)+1, 2^2^b(j-2)+1) = gcd(A000215(b(i-2)), A000215(b(j-2))) = 1 for 1 <= i < j, there is no n > 1 such that a(n) = n-1. Since b(i)-1 = 2^2^b(i-2)-1 divides b(j)-1 = 2^2^b(j-2)-1 for 1 <= i < j, a(n) = 1 if and only if n > 1 is a divisor of a number of the form b(i)-1, or if and only if n > 1 is a divisor of a Fermat number (A023394). - Jianing Song, May 16 2024

			a(5) = 1, as 2^x mod 5 is 1 for x being any even multiple of two and X = 2^(2^(2^...)) is an even multiple of two.

Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

Wayne VanWeerthuizen, Table of n, a(n) for n = 1..10000

Cf. A014221, A027763, A240162, A245971, A245972, A245973, A245974, A000010, A000079.

import Math.NumberTheory.Moduli (powerMod)
a245970 n = powerMod 2 (phi + a245970 phi) n
            where phi = a000010 n
-- Reinhard Zumkeller, Feb 01 2015

A:= proc(n)
     local phin,F,L,U;
     phin:= numtheory:-phi(n);
     if phin = 2^ilog2(phin) then
        F:= ifactors(n)[2];
        L:= map(t -> t[1]^t[2],F);
        U:= [seq(`if`(F[i][1]=2,0,1),i=1..nops(F))];
        chrem(U,L);
     else
        2 &^ A(phin) mod n
     fi
end proc:
seq(A(n), n=2 .. 100); # Robert Israel, Aug 19 2014

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[2, 2^n, n] (* 2^^(2^n) (mod n), in Knuth's up-arrow notation *); Array[f, 72]
(* Second program: *)
a[n_] := Module[{phin, F, L, U},
   phin = EulerPhi[n];
   If[phin == 2^Floor@Log2[phin],
      F = FactorInteger[n];
      L = Power @@@ F;
      U = Table[If[F[[i, 1]] == 2, 0, 1], {i, 1, Length[F]}];
      ChineseRemainder[U, L],
      (2^a[phin])~Mod~n]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 03 2023, after Robert Israel *)

a(n)=if(n<3, return(0)); my(e=valuation(n,2),k=n>>e); lift(chinese(Mod(2,k)^a(eulerphi(k)), Mod(0,2^e))) \\ Charles R Greathouse IV, Jul 29 2016

def tower2mod(n):
    if ( n <= 22 ):
        return 65536%n
    else:
        ep = euler_phi(n)
        return power_mod(2,ep+tower2mod(ep),n)

a(n) = 2^(A000010(n)+a(A000010(n))) mod n.
a(n) = 0 if n is a power of 2.
a(n) = (2^2) mod n, if n < 5.
a(n) = (2^(2^2)) mod n, if n < 11.
a(n) = (2^(2^(2^2))) mod n, if n < 23.
a(n) = (2^(2^(2^(2^2)))) mod n, if n < 47.
a(n) = (2^^k) mod n, if n < A027763(k), where ^^ is Knuth's double-arrow notation.
From Robert Israel, Aug 19 2014: (Start)
If gcd(m,n) = 1, then a(m*n) is the unique k in [0,...,m*n-1] with
k == a(n) mod n and k == a(m) mod m.
a(n) = 1 if n is a Fermat number.
a(n) = 2^a(A000010(n)) mod n if n is not in A003401.
(End)

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

A273950 Prime factors of generalized Fermat numbers of the form 12^(2^m) + 1 with m >= 0.

Original entry on oeis.org

5, 13, 17, 29, 89, 97, 233, 257, 769, 36097, 40961, 65537, 81281, 153953, 163841, 260753, 1724417, 4550657, 5767169, 8253953, 11304961, 13631489, 21495809, 69619841, 77651969, 147849217, 158334977, 159522817, 1711276033, 6528575489, 27286044673, 52613349377

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Primes p such that the multiplicative order of 12 (mod p) is a power of 2.

Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..44
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=12)
Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
OEIS Wiki, Generalized Fermat numbers

Cf. A023394, A072982, A152585, A268660, A268664, A273945 (base 3), A273946 (base 5), A273947 (base 6), A273948 (base 7), A273949 (base 11).

Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[12, #]] &]

A046052 Number of prime factors of Fermat number F(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5

Offset: 0

Eric W. Weisstein

F(12) has 6 known factors with C1133 remaining. [Updated by Walter Nissen, Apr 02 2010]
F(13) has 4 known factors with C2391 remaining.
F(14) has one known factor with C4880 remaining. [Updated by Matt C. Anderson, Feb 14 2010]
John Selfridge apparently conjectured that this sequence is not monotonic, so at some point a(n+1) < a(n). Related sequences such as A275377 and A275379 already exhibit such behavior. - Jeppe Stig Nielsen, Jun 08 2018
Factors are counted with multiplicity although it is unknown if all Fermat numbers are squarefree. - Jeppe Stig Nielsen, Jun 09 2018

W. Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
W. Keller, Summary of factoring status for Fermat numbersF(n)
PSI (The algorithm company), Fermat factor status [Broken link?]
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
Eric Weisstein's World of Mathematics, Fermat Number
Wikipedia, Selfridge's Conjecture about Fermat Numbers

Cf. A000215, A023394, A229850.

Array[PrimeOmega[2^(2^#) + 1] &, 9, 0] (* Michael De Vlieger, May 31 2022 *)

a(n)=bigomega(2^(2^n)+1) \\ Eric Chen, Jun 13 2018

a(n) = A001222(A000215(n)).

Name corrected by Arkadiusz Wesolowski, Oct 31 2011

A273945 Odd prime factors of generalized Fermat numbers of the form 3^(2^m) + 1 with m >= 0.

Original entry on oeis.org

5, 17, 41, 193, 257, 12289, 59393, 65537, 275201, 786433, 790529, 8972801, 13631489, 21523361, 134382593, 155189249, 448524289, 524455937, 847036417, 3221225473, 12348030977, 22320686081, 77309411329, 206158430209, 4638564679681, 6597069766657, 12079910333441

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p such that the multiplicative order of 3 (mod p) is a power of 2.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..35
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=3)
Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
OEIS Wiki, Generalized Fermat numbers
Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Cf. A023394, A059919, A072982, A268657, A268661, A273946 (base 5), A273947 (base 6), A273948 (base 7), A273949 (base 11), A273950 (base 12).

Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[3, #]] &]

cp's OEIS Frontend

A094358 Squarefree products of factors of Fermat numbers (A023394).

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A023395 Only Fermat number divisible by A023394(n) is 2^2^a(n) + 1.

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

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A344784 Decimal expansion of the sum of the reciprocals of the prime factors of Fermat numbers (A023394).

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A372891 Anti-elite primes (A128852) that are not prime factors of Fermat primes (A023394).

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A245970 Tower of 2's modulo n.

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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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