A129802 -id:A129802 - cp's OEIS frontend

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.

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)
}

A102742 Elite primes: a prime number p is called elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic residues mod p.

Original entry on oeis.org

3, 5, 7, 41, 15361, 23041, 26881, 61441, 87041, 163841, 544001, 604801, 6684673, 14172161, 159318017, 446960641, 1151139841, 3208642561, 38126223361, 108905103361, 171727482881, 318093312001, 443069456129, 912680550401, 1295536619521, 1825696645121, 2061584302081

Offset: 1

Tom Mueller, Feb 08 2005; extended Jun 16 2005

nonn

Křížek, Luca, Shparlinski, & Somer show that a(n) >> n log^2 n. - Charles R Greathouse IV, Jan 25 2017

Let d = 2^r*d' be the multiplicative order of 2 modulo p. Note that 2^2^s == 2^d == 1 (mod p), so p divides none of.

Dennis Martin, Table of n, a(n) for n = 1..29
Alexander Aigner, Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind, Monatsh. Math., Vol. 101 (1986), pp. 85-93; alternative link.
Alain Chaumont and Tom Mueller, All Elite Primes Up to 250 Billion, J. Integer Sequences, Vol. 9 (2006), Article 06.3.8.
Matthew Just, On upper bounds for the count of elite primes, arXiv:2102.00906 [math.NT], 2021.
Michal Křížek, Florian Luca, Igor E. Shparlinski, and Lawrence Somer, On the complexity of testing elite primes, Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.2, 5 pp.
Xiaoquin Li, Verifying Two Conjectures on Generalized Elite Primes, JIS 12 (2009) 09.4.7.
Dennis Martin, Elite Prime Search. [Broken link]
Dennis Martin, Elite Prime Search. [Cached copy, with permission of author]
Dennis Martin, Elite and Anti-Elite Prime Search Methodology [Cached copy, with permission of author]
Tom Müller, Searching for large elite primes, Experimental Mathematics, Vol. 15, Nol. 2 (2006), pp. 183-186.
Tom Muller and A. Reinhart, On generalized Elite Primes, Journal of Integer Sequences, Vol. 11 (2008), Article 08.3.1.
Tom Müller, On the Fermat Periods of Natural Numbers, Journal of Integer Sequences, Vol. 13 (2010), Article 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. A128852, A344785. Subsequence of A129802.

list_upto(N)={forprime(p=3,N,r=2^valuation(p-1,2); a=Mod(3,p); v=List(); k=0; while(1,listput(v,a); a=(a-1)^2+1; for(j=1,#v,if(v[j]==a,k=j;break(2)))); for(i=k,#v,znorder(v[i]) % r != 0 && next(2)); print1(p,", "))} \\ Slow, only for illustration, Jeppe Stig Nielsen, Jan 28 2020

isElite(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, 0) \\ Jianing Song, May 15 2024

Sum_{n>=1} 1/a(n) = A344785. - Amiram Eldar, May 30 2021

a(17) from Tom Mueller, Jul 20 2005
a(18)-a(21) from Tom Mueller, Apr 18 2006
6 further terms from Tom Mueller, Apr 16 2007

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

A351026 Possible bases b > 17 which can be used in Pepin's test to check the primality of any Fermat number greater than 5 only in the case when the base b is smaller than the tested number.

Original entry on oeis.org

51, 85, 102, 119, 170, 204, 238, 291, 340, 408, 459, 476, 485, 579, 582, 663, 679, 680, 697, 723, 765, 771, 816, 867, 918, 952, 965, 970, 1071, 1105, 1158, 1164, 1205, 1275, 1285, 1326, 1351, 1358, 1360, 1394, 1445, 1446, 1530, 1542, 1547, 1632, 1687, 1734, 1785

Offset: 1

Arkadiusz Wesolowski, Jan 29 2022

nonn

R. D. Carmichael, Fermat numbers F(n) = 2^(2^n) + 1, Amer. J. Math., 26 (1919), 137-146.
Eric Weisstein's World of Mathematics, Pepin's Test

Cf. A028730, A129802, A136804, A136806.

for(b=18, 1785, a=q=0; until(b-2<16^(2^a), a++; if(!(kronecker(b, 16^(2^(a-1))+1)==-1), q=1; break)); if(q==1, 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, ", ")))));

A positive integer b belongs to this sequence if and only if the Jacobi symbol J(b,F(m)) has value 0 or 1 for some 5 < F(m) < b, and J(b,F(m)) = 1 only for a finite number of Fermat numbers F(m) = 2^(2^m) + 1.

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)
}

A352536 Terms in A351026 that are not divisible by any Fermat number > 5.

Original entry on oeis.org

291, 485, 579, 582, 679, 723, 965, 970, 1158, 1164, 1205, 1351, 1358, 1446, 1687, 1923, 1930, 1940, 2019, 2316, 2328, 2410, 2619, 2702, 2716, 2892, 3205, 3365, 3374, 3783, 3846, 3860, 3880, 3977, 4038, 4365, 4487, 4632, 4656, 4711, 4820, 5211, 5238, 5404, 5432

Offset: 1

Arkadiusz Wesolowski, Mar 22 2022

nonn

Not every term of A129802 that is not divisible by any Fermat number > 5 can be used for the base "5" in Pepin's test (if a tested Fermat number is smaller than the base).

Cf. A129802, A351026.

cp's OEIS Frontend

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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A102742 Elite primes: a prime number p is called elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic residues mod p.

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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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A351026 Possible bases b > 17 which can be used in Pepin's test to check the primality of any Fermat number greater than 5 only in the case when the base b is smaller than the tested number.

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

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A352536 Terms in A351026 that are not divisible by any Fermat number > 5.

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