A001583 -id:A001583 - cp's OEIS frontend

A047272 Primes p such that p == 1 (mod 4) and Fibonacci((p-1)/4) is divisible by p.

61, 89, 109, 149, 269, 389, 401, 421, 521, 661, 701, 761, 769, 809, 821, 829, 1009, 1049, 1069, 1181, 1249, 1289, 1301, 1361, 1381, 1409, 1429, 1601, 1721, 1901, 1949, 2081, 2089, 2141, 2221, 2269, 2309, 2341, 2381, 2441, 2549, 2729, 2741, 2749, 2789

Offset: 1

N. J. A. Sloane

There are typos in Theorems 1 and 2 of Lehmer's paper. The condition a == 1 (mod 4) should be a == 1 (mod 2). - Eric Rowland, May 20 2018

E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138.
E. Lehmer, On the quadratic character of the Fibonacci root (annotated scanned copy)

For nonprimes see A091982.

Select[Prime[Range[400]], Function[p, Mod[p, 4] == 1 && Divisible[Fibonacci[(p - 1)/4], p]]] (* Eric Rowland, May 20 2018 *)

More terms from James Sellers, Jan 25 2000

New Name from Eric Rowland, May 20 2018

A047651 Minimal positive value of tau modulo primes in A047650.

Original entry on oeis.org

6, 10, 23, 14, 82, 144, 112, 22, 122, 100, 173, 171, 92, 339, 343, 228, 383, 458, 326, 460, 407, 485, 405, 157, 83, 125, 530, 606, 176, 136, 601, 42, 321, 236, 364, 842, 46, 914, 612, 676, 380, 535, 335, 393, 54, 123, 1209, 1298, 812, 1196, 1371, 252, 314, 851, 524, 1157

Offset: 1

N. J. A. Sloane

Since sqrt(5) mod p is determined up to a sign, there are two values that tau mod p can take.

Minimal positive solution to the equation x^2 - x - 1 = 0 modulo p for primes p in A047650.

E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138.
E. Lehmer, On the quadratic character of the Fibonacci root (annotated scanned copy)

Cf. A047650.

Edited and extended by Max Alekseyev, Feb 06 2010

A168172 Least prime p == -1 (mod n) that divides Fibonacci((p+1)/n), or 0 if no such prime exists.

Original entry on oeis.org

2, 13, 47, 0, 0, 113, 307, 0, 233, 0, 967, 0, 2417, 797, 0, 0, 1087, 233, 5737, 0, 5417, 5653, 1103, 0, 0, 2417, 4373, 0, 6263, 0, 25357, 0, 3167, 42533, 0, 0, 4513, 5737, 2417, 0, 61417, 5417, 32507, 0, 0, 36017, 1597, 0, 97607, 0, 27947, 0, 42293, 4373, 0, 0

Offset: 1

M. F. Hasler, Nov 28 2009

nonn

Max Alekseyev has proved (cf. link) that a(n)=0 if n is a multiple of 4 or 5; for all other n, a prime a(n) with the required property seems to exist.

Max Alekseyev, Re: Primes p = nk-1 dividing Fibonacci( k ), SeqFan mailing list, Nov. 2009.

A168172(n) = n%4 && n%5 && forstep(p=n-1,1e9,n, isprime(p) || next; fibonacci((p+1)/n)%p || return(p))

A271263 Septic artiads (A270800) congruent to 1 mod 98 and for which 2 is a 7th power residue.

Original entry on oeis.org

874651, 941879, 1074277, 1080451, 1396697, 2024387, 2546237, 2807603, 3267419, 3324847, 3436273, 3465673, 3851009, 4150301, 4219979, 4367567, 4651963, 4762507, 5173813, 5308759, 5398919, 5474477, 5552191, 5710363, 6248579, 6391267, 6575507, 6627251, 6791107

Offset: 1

Eric M. Schmidt, Apr 03 2016

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..100
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131. See p. 128, Theorem 7.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

Cf. A001583, A270800.

def isa(n) :
    if not (n % 98 == 1 and is_prime(n)) : return False
    R. = PolynomialRing(GF(n))
    return 2.powermod((n-1)//7, n) == 1 and all(r[0]^((n-1)//7) == 1 for r in (t^3 + t^2 - 2*t - 1).roots())

A271264 Septic artiads (A270800) congruent to 1 mod 98 and for which 7 is a 7th power residue.

Original entry on oeis.org

876611, 1163653, 1471079, 1608769, 2367289, 2368759, 2538103, 2564857, 2621501, 2693629, 2774381, 3120713, 3495269, 3498797, 3636781, 3974881, 4240853, 4376681, 4571309, 4654217, 4702433, 4867171, 5047883, 5066993, 5629121, 5644213, 5760343, 5779649, 6262397

Offset: 1

Eric M. Schmidt, Apr 03 2016

nonn

Septic hyperartiads (A270801) congruent to 1 mod 98.

Eric M. Schmidt, Table of n, a(n) for n = 1..100
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131. See p. 128-129, Theorem 8.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

Cf. A001583, A270800, A270801.

def isa(n) :
    if not (n % 98 == 1 and is_prime(n)) : return False
    R. = PolynomialRing(GF(n))
    return 7.powermod((n-1)//7, n) == 1 and all(r[0]^((n-1)//7) == 1 for r in (t^3 + t^2 - 2*t - 1).roots())

A271265 Primes congruent to 1 mod 14 represented by x^2 + 343y^2.

Original entry on oeis.org

379, 743, 827, 1373, 1499, 1597, 2213, 2647, 2843, 3221, 3571, 4243, 4397, 4621, 5657, 6133, 6217, 6329, 6427, 8443, 8513, 8597, 8737, 8807, 8863, 9059, 9871, 10529, 10781, 11159, 11173, 11551, 11579, 12377, 12517, 12671, 12713, 13693, 13903, 14029, 14197

Offset: 1

Eric M. Schmidt, Apr 03 2016

nonn

Includes the septic artiads (A270800).

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131. See p. 129-130, Theorems 9 and 10.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

Cf. A270800.

A271171 Erroneous version of A270798.

Original entry on oeis.org

5081, 5591, 6211, 6271, 8581, 8861, 9011, 9661

Offset: 1

dead

The correct first term is 5281, not 5081 as given in [Lehmer].

E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

cp's OEIS Frontend

A047272 Primes p such that p == 1 (mod 4) and Fibonacci((p-1)/4) is divisible by p.

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A047651 Minimal positive value of tau modulo primes in A047650.

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A168172 Least prime p == -1 (mod n) that divides Fibonacci((p+1)/n), or 0 if no such prime exists.

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PARI

A271263 Septic artiads (A270800) congruent to 1 mod 98 and for which 2 is a 7th power residue.

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Sage

A271264 Septic artiads (A270800) congruent to 1 mod 98 and for which 7 is a 7th power residue.

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Programs

Sage

A271265 Primes congruent to 1 mod 14 represented by x^2 + 343y^2.

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A271171 Erroneous version of A270798.

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