A270800 -id:A270800 - cp's OEIS frontend

A270801 Septic hyperartiads: septic artiads (A270800) for which 7 is a 7th power residue.

665897, 741413, 794207, 859601, 876611, 892627, 980911, 1102249, 1116977, 1123879, 1129213, 1163653, 1228543, 1237139, 1393771, 1434553, 1453019, 1471079, 1513163, 1570577, 1588133, 1608769, 1638211, 1638743, 1645253, 1670887, 1702933, 1704137, 1785337

Offset: 1

N. J. A. Sloane, Apr 01 2016, typo corrected Apr 02 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.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

Cf. A001583, A270800.

def is_septic_hyperartiad(n) :
    if not (n % 14 == 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())
# Eric M. Schmidt, Apr 02 2016

Definition corrected by and more terms from Eric M. Schmidt, Apr 02 2016

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())

A271247 Erroneous version of A270800.

Original entry on oeis.org

14197, 21617, 25801, 24977, 25999

Offset: 1

dead

The correct third term is 23801, not 25801 as given in [Lehmer].

E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131.

A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

Original entry on oeis.org

211, 281, 421, 461, 521, 691, 881, 991, 1031, 1151, 1511, 1601, 1871, 1951, 2221, 2591, 3001, 3251, 3571, 3851, 4021, 4391, 4441, 4481, 4621, 4651, 4691, 4751, 4871, 5081, 5281, 5381, 5531, 5591, 5641, 5801, 5881, 6011, 6101, 6211, 6271, 6491, 6841

Offset: 1

N. J. A. Sloane

From A.H.M. Smeets, Nov 15 2023: (Start)
Mean gap size between two consecutive terms at p: ~ 20*log(p) (see E. Lehmer).
In E. Lehmer, Artiads characterized, she counted in the table on p. 122 the primes p for which p == 1 (mod 5) instead of all primes. As a result, in the corollary on p. 121, the 20% becomes 5% (or 1/20 instead of 1/5). (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..24903 (first 1000 terms from T. D. Noe)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
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]
E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138 (annotated scanned copy).
Michael J. Mossinghoff and Christopher Pinner, Prime power order circulant determinants, arXiv:2205.12439 [math.NT], 2022. See Type 2 primes on p. 3.
H. W. Lloyd Tanner, On the Binomial Equation x^p-1=0: Quinquisection, Proc. London Math. Soc., 18 (1886-1887), 214-234.
H. W. Lloyd Tanner, On Complex Primes formed with the Fifth Roots of Unity, Proc. London Math. Soc., 24 (1892-1893), 223-262.

Cf. A047650, A000045, A024894, subsequence of A030430.

See also A270798 (a subsequence), A270800.

a001583 n = a001583_list !! (n-1)
a001583_list = filter
   (\p -> mod (a000045 $ div (p - 1) 5) p == 0) a030430_list
-- Reinhard Zumkeller, Aug 15 2013

Select[ Prime[ Range[1000]], Mod[#, 5] == 1 && Divisible[ Fibonacci[(# - 1)/5], #] &] (* Jean-François Alcover, Jun 22 2012 *)

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
list(lim)=my(v=List()); forprime(p=11,lim, if(p%5==1 && fibmod(p\5,p)==0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 06 2017

From A.H.M. Smeets, Nov 15 2023: (Start)
Equals {prime(m): A296240(m) == 0 (mod 5)}.
a(n) ~ prime(20*n). (End)

More terms from James Sellers, Jan 25 2000

Edited by N. J. A. Sloane, Apr 01 2016

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.

cp's OEIS Frontend

A270801 Septic hyperartiads: septic artiads (A270800) for which 7 is a 7th power residue.

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

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

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A271247 Erroneous version of A270800.

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A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

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

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