A102742 -id:A102742 - cp's OEIS frontend

A344785 Decimal expansion of the sum of the reciprocals of the elite primes (A102742).

7, 0, 0, 7, 6, 4, 0, 1, 1, 5

Offset: 0

Amiram Eldar, May 28 2021

Křížek et al. (2002) proved that this series is convergent.

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

			0.7007640115...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, Section 1.37, p. 247.

Steven R. Finch, Fermat Numbers and Elite Primes, preprint, 2013.
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. A102742.

Equals Sum_{k>=1} 1/A102742(k).

A372895 Squarefree terms of A129802 whose prime factors are neither elite (A102742) nor anti-elite (A128852), where A129802 is the possible bases for Pepin's primality test for Fermat numbers.

Original entry on oeis.org

551, 1387, 2147, 8119, 8227, 8531, 10483, 21907, 29261, 29543, 30229, 52909, 58133, 65683, 73657, 81257, 81797, 84491, 89053, 89281, 97907, 114017, 184987, 187891, 227557, 228997, 238111, 263017, 369721, 405631, 436897, 450607, 453041, 468541, 472967, 498817, 521327, 641297, 732127, 736003, 810179, 930677

Offset: 1

Jianing Song, May 15 2024

nonn

By construction, A129802 is the disjoint union of the two following sets of numbers: (a) products of a square, some distinct anti-elite primes, an even number of elite-primes and a term here; (b) products of a square, some distinct anti-elite primes, an odd number of elite-primes and a term in A372896.

Cf. A129802, A102742, A128852, A372894, A372896.

isA372895(n) = {
if(n%2 && issquarefree(n) && isA129802(n), my(f = factor(n)~[1,]); \\ See A129802 for its program
for(i=1, #f, my(p=f[i], d = znorder(Mod(2, p)), StartPoint = valuation(d, 2), LengthTest = znorder(Mod(2, d >> StartPoint)), flag = 0); \\ To check if p = f[i] is an elite prime or an anti-elite prime, it suffices to check (2^2^i + 1) modulo p for StartPoint <= i <= StartPoint + LengthTest - 1; see A129802 or A372894
for(j = StartPoint+1, StartPoint + LengthTest - 1, if(issquare(Mod(2, p)^2^j + 1) != issquare(Mod(2, p)^2^StartPoint + 1), flag = 1; break())); if(flag == 0, return(0))); 1, 0)
}

A372896 Squarefree terms of A372894 whose prime factors are neither elite (A102742) nor anti-elite (A128852).

Original entry on oeis.org

1, 341, 671, 1891, 2117, 3277, 4033, 5461, 8249, 12557, 13021, 14531, 19171, 24811, 31609, 32777, 33437, 40951, 46139, 48929, 49981, 50737, 73279, 80581, 84169, 100253, 116143, 130289, 135923, 136271, 149437, 175577, 179783, 194417, 252361, 272491, 342151, 343027, 376169, 390641

Offset: 1

Jianing Song, May 15 2024

nonn

By construction, A372894 is the disjoint union of the two following sets of numbers: (a) products of a square, some distinct anti-elite primes, an even number of elite-primes and a term here; (b) products of a square, some distinct anti-elite primes, an odd number of elite-primes and a term in A372895.

Cf. A372894, A102742, A128852, A129802, A372895.

isA372896(n) = {
if(n%2 && issquarefree(n) && isA372894(n), if(n==1, return(1)); my(f = factor(n)~[1,]); \\ See A372894 for its program
for(i=1, #f, my(p=f[i], d = znorder(Mod(2, p)), StartPoint = valuation(d, 2), LengthTest = znorder(Mod(2, d >> StartPoint)), flag = 0); \\ To check if p = f[i] is an elite prime or an anti-elite prime, it suffices to check (2^2^i + 1) modulo p for StartPoint <= i <= StartPoint + LengthTest - 1; see A129802 or A372894
for(j = StartPoint+1, StartPoint + LengthTest - 1, if(issquare(Mod(2, p)^2^j + 1) != issquare(Mod(2, p)^2^StartPoint + 1), flag = 1; break())); if(flag == 0, return(0))); 1, 0)
}

A346542 Numbers k such that 5*2^k + 1 is an elite prime (A102742).

Original entry on oeis.org

3, 15, 55, 26607, 209787, 819739

Offset: 1

Arkadiusz Wesolowski, Sep 16 2021

An integer k is in this sequence if and only if there is no solution to the congruence x^2 == 2^(2^k) + 1 (mod p), where p is a prime of the form 5*2^k + 1.

a(7) > 9*10^6.

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

Subsequence of A002254.

Cf. A102742.

isok(k)=my(p=5*2^k+1); k>2 && Mod(k, 2)==1 && Mod(3, p)^((p-1)/2)+1==0 && kronecker(lift(Mod(2, p)^2^k)+1, p)==-1;

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

A129802 Possible bases for Pepin's primality test for Fermat numbers.

Original entry on oeis.org

3, 5, 6, 7, 10, 12, 14, 20, 24, 27, 28, 39, 40, 41, 45, 48, 51, 54, 56, 63, 65, 75, 78, 80, 82, 85, 90, 91, 96, 102, 105, 108, 112, 119, 125, 126, 130, 147, 150, 156, 160, 164, 170, 175, 180, 182, 192, 204, 210, 216, 224, 238, 243, 245, 250, 252, 260, 291, 294, 300

Offset: 1

Max Alekseyev, Jun 14 2007, corrected Dec 29 2007. Thanks to Ant King for pointing out an error in the earlier version of this sequence

nonn

Prime elements of this sequence are given by A102742.
From Jianing Song, May 15 2024: (Start)
Let m be an odd number and ord(2,m) = 2^r*d be the multiplicative order of 2 modulo m, where d is odd, then 2^2^n + 1 is congruent to one of 2^2^r + 1, 2^2^(r+1) + 1, ..., 2^2^(r+ord(2,d)-1) + 1 modulo m, so it suffices to check these ord(2,d) numbers.
Note that if m > 1, then m does not divide 2^2^n + 1 for n >= r, otherwise we would have 2^(2^n*d) = (2^ord(2,m))^2^(n-r) == 1 (mod m) and 2^(2^n*d) = (2^2^n)^d == (-1)^d == -1 (mod m). As a result, m is a term if and only if the Jacobi symbol ((2^2^n + 1)/m) is equal to -1 for m = r, r+1, ..., r+ord(2,d)-1.
By definition, a squarefree number that is a product of elite primes (A102742) or anti-elite primes (A128852) is a term if and only if its number of elite factors is odd. But a squarefree term can have factors that are neither elite nor anti-elite, the smallest being 551 = 19*29. (End)

			For n >= 2, we have 2^2^n + 1 == 170, 461, 17, 257, 519, 539 (mod 551) respectively for n == 0, 1, 2, 3, 4, 5 (mod 6). As we have (170/551) = (461/551) = (17/551) = (257/551) = (519/551) = (539/551) = -1, 551 is a term. - _Jianing Song_, May 19 2024

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pepin's Test.

Cf. A000215, A019434, A060377, A102742, A128852, A372894.

{ isPepin(n) = local(s,S=Set(),t); n\=2^valuation(n,2); s=Mod(3,n); while( !setsearch(S,s), S=setunion(S,[s]); s=(s-1)^2+1); t=s; until( t==s, if( kronecker(lift(t),n)==1, return(0)); t=(t-1)^2+1);1 }
for(n=2,1000,if(isPepin(n),print1(n,", ")))

for(b=2, 300, k=b/2^valuation(b, 2); if(k>1, i=logint(k, 2); m=Mod(2, k); z=znorder(m); e=znorder(Mod(2, z/2^valuation(z, 2))); t=0; for(c=1, e, if(kronecker(lift(m^2^(i+c))+1, k)==-1, t++, break)); if(t==e, print1(b, ", ")))); \\ Arkadiusz Wesolowski, Sep 22 2021

isA129802(n) = n = (n >> valuation(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(kronecker(lift(Mod(2, n)^2^i + 1), n) == 1, return(0))); 1 \\ Jianing Song, May 19 2024

A positive integer 2^k*m, where m is odd and k >= 0, belongs to this sequence iff the Jacobi symbol (F_n/m) = 1 for only a finite number of Fermat numbers F_n = A000215(n).

A372894 A positive integer 2^k*m, where m is odd and k >= 0, belongs to this sequence iff the Jacobi symbol (F_n/m) = -1 for only a finite number of Fermat numbers F_n = A000215(n).

Original entry on oeis.org

1, 2, 4, 8, 9, 13, 15, 16, 17, 18, 21, 25, 26, 30, 32, 34, 35, 36, 42, 49, 50, 52, 60, 64, 68, 70, 72, 81, 84, 97, 98, 100, 104, 117, 120, 121, 123, 128, 135, 136, 140, 144, 153, 162, 168, 169, 189, 193, 194, 195, 196, 200, 205, 208, 221, 225, 234, 240, 241, 242, 246, 255, 256, 257, 270, 272, 273, 280, 287, 288, 289

Offset: 1

Jianing Song, May 15 2024

nonn

Can be seen as the opposite of A129802.
Let m be an odd number and ord(2,m) = 2^r*d be the multiplicative order of 2 modulo m, where d is odd, then 2^2^n + 1 is congruent to one of 2^2^r + 1, 2^2^(r+1) + 1, ..., 2^2^(r+ord(2,d)-1) + 1 modulo m, so it suffices to check these ord(2,d) numbers.
Note that if m > 1, then m does not divide 2^2^n + 1 for n >= r, otherwise we would have 2^(2^n*d) = (2^ord(2,m))^2^(n-r) == 1 (mod m) and 2^(2^n*d) = (2^2^n)^d == (-1)^d == -1 (mod m). As a result, m is a term if and only if the Jacobi symbol ((2^2^n + 1)/m) is equal to 1 for m = r, r+1, ..., r+ord(2,d)-1.
By definition, a squarefree number that is a product of elite primes (A102742) or anti-elite primes (A128852) is a term if and only if its number of elite factors is even. But a squarefree term can have factors that are neither elite nor anti-elite, the smallest being 341 = 11*31.
Contains divisors of Fermat numbers >= 17 (A307843 \ {3,5}) since they are products of elite primes.

			For n >= 1, we have 2^2^n + 1 == 65, 5, 17, 257 (mod 341) respectively for n == 0, 1, 2, 3 (mod 4). As we have (65/341) = (5/341) = (17/341) = (257/341) = 1, 341 is a term.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A000215, A102742, A129802, A307843.

Prime elements of this sequence are given by A128852.

isA372894(n) = n = (n >> valuation(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(kronecker(lift(Mod(2, n)^2^i + 1), n) == -1, return(0))); 1

A348062 Primes p such that the length of the (eventual) period of the sequence {2^(2^k) mod p: k >= 0} is odd.

Original entry on oeis.org

2, 3, 5, 17, 29, 43, 47, 113, 127, 179, 197, 257, 277, 283, 293, 317, 383, 439, 449, 467, 479, 509, 569, 641, 659, 719, 797, 863, 1013, 1069, 1289, 1373, 1399, 1427, 1439, 1487, 1579, 1627, 1657, 1753, 1823, 1913, 1933, 1949, 2063, 2203, 2207, 2213, 2273, 2339, 2351

Offset: 1

Arkadiusz Wesolowski, Sep 26 2021

nonn

Of these numbers only 3 and 5 are elite primes (A102742). (Aigner)

Every prime of the form A036259(n)*2^m + 1, with m, n >= 1, is in this sequence.

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

Supersequence of A023394.

Cf. A102742 (elite primes), A256607.

L=List([2]); forprime(p=3, 2351, z=znorder(Mod(2, p)); if(znorder(Mod(2, z/2^valuation(z, 2)))%2, listput(L, p))); Vec(L)

cp's OEIS Frontend

A344785 Decimal expansion of the sum of the reciprocals of the elite primes (A102742).

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A372895 Squarefree terms of A129802 whose prime factors are neither elite (A102742) nor anti-elite (A128852), where A129802 is the possible bases for Pepin's primality test for Fermat numbers.

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A372896 Squarefree terms of A372894 whose prime factors are neither elite (A102742) nor anti-elite (A128852).

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A346542 Numbers k such that 5*2^k + 1 is an elite prime (A102742).

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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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A129802 Possible bases for Pepin's primality test for Fermat numbers.

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A372894 A positive integer 2^k*m, where m is odd and k >= 0, belongs to this sequence iff the Jacobi symbol (F_n/m) = -1 for only a finite number of Fermat numbers F_n = A000215(n).

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A348062 Primes p such that the length of the (eventual) period of the sequence {2^(2^k) mod p: k >= 0} is odd.

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