A023394 -id:A023394 - cp's OEIS frontend

A273946 Odd prime factors of generalized Fermat numbers of the form 5^(2^m) + 1 with m >= 0.

3, 13, 17, 257, 313, 641, 769, 2593, 11489, 19457, 65537, 163841, 786433, 1503233, 1655809, 7340033, 14155777, 18395137, 23606273, 29423041, 39714817, 75068993, 167772161, 2483027969, 4643094529, 6616514561, 47148957697, 241931001601, 2748779069441

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p such that the multiplicative order of 5 (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..37
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=5)
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, A199591, A268658, A268662, A273945 (base 3), A273947 (base 6), A273948 (base 7), A273949 (base 11), A273950 (base 12).

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

A273947 Prime factors of generalized Fermat numbers of the form 6^(2^m) + 1 with m >= 0.

Original entry on oeis.org

7, 17, 37, 257, 353, 1297, 1697, 2753, 18433, 65537, 80897, 98801, 145601, 763649, 3360769, 4709377, 13631489, 50307329, 376037377, 2483027969, 3191106049, 4926056449, 51808043009, 152605556737, 916326983681, 1268357529601, 6597069766657, 40711978221569

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Primes p other than 5 such that the multiplicative order of 6 (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..34
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=6)
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, Some factors of the numbers G_n=6^(2^n)+1 and H_n=10^(2^n)+1, Math. Comp. 23 (1969), no. 106, pp. 413-415.

Cf. A023394, A072982, A078303, A268663, A273945 (base 3), A273946 (base 5), A273948 (base 7), A273949 (base 11), A273950 (base 12).

Select[Prime@Range[4, 10^5], IntegerQ@Log[2, MultiplicativeOrder[6, #]] &]

A273948 Odd prime factors of generalized Fermat numbers of the form 7^(2^m) + 1 with m >= 0.

Original entry on oeis.org

5, 17, 257, 353, 769, 1201, 12289, 13313, 35969, 65537, 114689, 163841, 169553, 7699649, 9379841, 11886593, 28667393, 64749569, 70254593, 134818753, 197231873, 4643094529, 19847446529, 47072139617, 206158430209, 452850614273, 531968664833, 943558259713

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p other than 3 such that the multiplicative order of 7 (mod p) is a power of 2.
From Robert Israel, Jun 16 2016: (Start)
If p is in the sequence, then for each m either p | 7^(2^k)+1 for some k < m or 2^m | p-1.  Thus all members except 5, 17, 353, 1201, 169553, 7699649, 134818753, 47072139617 are congruent to 1 mod 2^7.
The intersection of this sequence and A019337 is A019434 minus {3}. (End)

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..34
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.
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, A078304, A273945 (base 3), A273946 (base 5), A273947 (base 6), A273949 (base 11), A273950 (base 12).

filter:= proc(t)
  if not isprime(t) then return false fi;
  7 &^ (2^padic:-ordp(t-1,2)) mod t = 1
end proc:
select(filter, [seq(i,i=5..10^6,2)]); # Robert Israel, Jun 16 2016

Select[Prime@Range[3, 10^5], IntegerQ@Log[2, MultiplicativeOrder[7, #]] &]

A273949 Odd prime factors of generalized Fermat numbers of the form 11^(2^m) + 1 with m >= 0.

Original entry on oeis.org

3, 17, 61, 193, 257, 7321, 15361, 51329, 65537, 163841, 6304673, 15190529, 70254593, 1691123713, 1760464897, 3221225473, 3489660929, 4696846849, 6874464257, 53401878529, 111489577217, 149300051969, 184683593729, 206158430209, 447600088289, 1819992391681

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p other than 5 such that the multiplicative order of 11 (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..31
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.
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, A199592, A273945 (base 3), A273946 (base 5), A273947 (base 6), A273948 (base 7), A273950 (base 12).

Delete[Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[11, #]] &], 2]

A128852 Anti-elite primes: a prime number p is called anti-elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic non-residues mod p.

Original entry on oeis.org

2, 13, 17, 97, 193, 241, 257, 641, 673, 769, 2689, 5953, 8929, 12289, 40961, 49921, 61681, 65537, 101377, 114689, 274177, 286721, 319489, 414721, 417793, 550801, 786433, 974849, 1130641, 1376257, 1489153, 1810433, 2424833, 3602561, 6700417

Offset: 1

Tom Mueller, Apr 16 2007

nonn

There are infinitely many anti-elite primes.

			Let F_r:=2^(2^r)+1 = r-th Fermat number. Then a(2)=13 because for all r>1 we have F_r == 4 (mod 13) if r is even, resp. F_r == 10 (mod 13) if r is odd. Notice that 4 and 10 are quadratic residues modulo 13.

Alexander Aigner; Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind. Monatsh. Math. 101 (1986), pp. 85-93

Dennis Martin, Table of n, a(n) for n = 1..101
M. Krizek, F. Luca, I. E. Shparlinski, L. Somer, On the complexity of testing elite primes, J. Int. Seq. 14 (2011) # 11.1.2
Dennis Martin, Anti-Elite Prime Search
Dennis Martin, Anti-Elite Prime Search [Cached copy, with permission of author]
Tom Müller, On Anti-Elite Prime Numbers, J. Integer Sequences, Vol. 10 (2007), Article 07.9.4.
Tom Müller, On the Fermat Periods of Natural Numbers, J. Int. Seq. 13 (2010) # 10.9.5.
Tom Müller, On the Exponents of Non-Trivial Divisors of Odd Numbers and a Generalization of Proth's Primality Theorem, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.7.

Cf. A102742.

Contains all Fermat prime factors of Fermat numbers (A023394) that are greater than 5.

isAntiElite(n) = if(isprime(n) && n > 2, my(d = znorder(Mod(2,n)), 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) \\ Jianing Song, May 15 2024

A228539 Rows of binary Walsh matrices interpreted as reverse binary numbers.

Original entry on oeis.org

0, 0, 2, 0, 10, 12, 6, 0, 170, 204, 102, 240, 90, 60, 150, 0, 43690, 52428, 26214, 61680, 23130, 15420, 38550, 65280, 21930, 13260, 39270, 4080, 42330, 49980, 27030, 0, 2863311530, 3435973836, 1717986918, 4042322160, 1515870810, 1010580540

Offset: 0

Tilman Piesk, Aug 24 2013

T(n,k) is row k of the binary Walsh matrix of size 2^n read as reverse binary number. The left digit is always 0, so all entries are even.
Most of these numbers are divisible by Fermat numbers (A000215): All entries in all rows beginning with row n are divisible by F_(n-1), except the entries 2^(n-1)...2^n-1. (This is the same in A228540.)
Divisibility by Fermat numbers:
All entries are divisible by F_0 = 3, except those with k = 1.
All entries in rows n >= 3 are divisible by F_2 = 17, except those with k = 4..7.

			Binary Walsh matrix of size 4 and row 2 of the triangle:
0 0 0 0         0
0 1 0 1        10
0 0 1 1        12
0 1 1 0         6
Triangle starts:
   k  =  0     1     2     3     4     5     6     7     8     9    10    11 ...
n
0        0
1        0     2
2        0    10    12     6
3        0   170   204   102   240    90    60   150
4        0 43690 52428 26214 61680 23130 15420 38550 65280 21930 13260 39270 ...

Tilman Piesk, Rows 0..8 of the triangle, flattened
Tilman Piesk, Prime factorizations
Tilman Piesk, Binary Walsh matrix of size 256

Cf. A228540 (the same for the negated binary Walsh matrix).

Cf. A000215 (Fermat numbers), A023394 (Prime factors of Fermat numbers).

T(n,k) + A228540(n,k) = 2^2^n - 1

T(n,2^n-1) = A122570(n+1)

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

A228540 Rows of negated binary Walsh matrices interpreted as reverse binary numbers.

Original entry on oeis.org

1, 3, 1, 15, 5, 3, 9, 255, 85, 51, 153, 15, 165, 195, 105, 65535, 21845, 13107, 39321, 3855, 42405, 50115, 26985, 255, 43605, 52275, 26265, 61455, 23205, 15555, 38505, 4294967295, 1431655765, 858993459, 2576980377, 252645135, 2779096485, 3284386755

Offset: 0

Tilman Piesk, Aug 24 2013

T(n,k) is row k of the negated binary Walsh matrix of size 2^n read as reverse binary number. The left digit is always 1, so all entries are odd.
Most of these numbers are divisible by Fermat numbers (A000215): All entries in all rows beginning with row n are divisible by F_(n-1), except the entries 2^(n-1)...2^n-1. (This is the same in A228539.)
Divisibility by Fermat numbers:
All entries in rows n >= 1 are divisible by F_0 =  3, except those with k = 1.
All entries in rows n >= 3 are divisible by F_2 = 17, except those with k = 4..7.

			Negated binary Walsh matrix of size 4 and row 2 of the triangle:
1 1 1 1        15
1 0 1 0         5
1 1 0 0         3
1 0 0 1         9
Triangle starts:
      k  =  0     1     2     3    4     5     6     7   8     9    10    11 ...
n
0           1
1           3     1
2          15     5     3     9
3         255    85    51   153   15   165   195   105
4       65535 21845 13107 39321 3855 42405 50115 26985 255 43605 52275 26265 ...

Tilman Piesk, Rows 0..8 of the triangle, flattened
Tilman Piesk, Prime factorizations
Tilman Piesk, Negated binary Walsh matrix of size 256

A228539 (the same for the binary Walsh matrix, not negated)
A197818 (antidiagonals of the negated binary Walsh matrix converted to decimal).
A000215 (Fermat numbers), A023394 (Prime factors of Fermat numbers).

T(n,k) + A228539(n,k) = 2^2^n - 1
T(n,0) = A051179(n)
T(n,2^n-1) = A122569(n+1)
A211344(n,k) = T(n,2^(n-k))

A228846 Largest m such that (2k+1)*2^m + 1 is a prime factor of the Fermat number 2^(2^n) + 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 7, 8, 9, 11, 16, 14, 14

Offset: 0

Arkadiusz Wesolowski, Sep 05 2013

a(n) >= n + 2 for n >= 2.
a(n) = A228845(n) if 2^(2^n) + 1 is prime or semiprime.
a(n) = max (A007814(p_i-1)), where p_i are the prime factors of 2^(2^n)+1. - Ralf Stephan, Sep 09 2013
For n >= 2, a(n) >= n + 3 if A046052(n) is an odd number. - Arkadiusz Wesolowski, Aug 10 2021

			F(5) = 641*6700417 and max(A007814(640),A007814(6700416))=7, so a(5)=7.

Wilfrid Keller, Fermat factoring status
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A019434, A023394, A046052, A228845.

A229850 Number of prime factors congruent to 1 mod 3 that divide the Fermat number 2^(2^n) + 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 3, 2

Offset: 0

Arkadiusz Wesolowski, Oct 01 2013

a(n) < A046052(n) because all Fermat numbers greater than 3 are equal to 2 (mod 3).
a(n) = 1 if A046052(n) = 2.
If A046052(n) = 3, then a(n) = 0 or 2.
a(n) <= A228846(n) - n - 1 for n = 0 to 11.

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, pp. 61-63, 65-66.

Wilfrid Keller, Fermat factoring status
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A002476, A023394, A046052, A229854.

cp's OEIS Frontend

A273946 Odd prime factors of generalized Fermat numbers of the form 5^(2^m) + 1 with m >= 0.

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A273947 Prime factors of generalized Fermat numbers of the form 6^(2^m) + 1 with m >= 0.

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A273948 Odd prime factors of generalized Fermat numbers of the form 7^(2^m) + 1 with m >= 0.

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A273949 Odd prime factors of generalized Fermat numbers of the form 11^(2^m) + 1 with m >= 0.

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A128852 Anti-elite primes: a prime number p is called anti-elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic non-residues mod p.

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PARI

A228539 Rows of binary Walsh matrices interpreted as reverse binary numbers.

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Formula

A307843 Divisors of Fermat numbers.

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A228540 Rows of negated binary Walsh matrices interpreted as reverse binary numbers.

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