A001583 -id:A001583 - cp's OEIS frontend

A270799 Artiads (A001583) congruent to 1 mod 50 and having 2 as a quintic residue.

3251, 4751, 14251, 17401, 21401, 27551, 32051, 32251, 36151, 36451, 42451, 51001, 52501, 54101, 55001, 56501, 59051, 60101, 61051, 83401, 104801, 113051, 116101, 119851, 121351, 170701, 174901, 178501, 178601, 179051, 182101, 185951, 190301, 202751, 213901

Offset: 1

N. J. A. Sloane, Apr 01 2016

nonn

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 page 123.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

Cf. A001583.

def isa(n) : return n % 50 == 1 and is_prime(n) and 2.powermod((n-1)//5, n) == 1 and fibonacci((n - 1)//5) % n == 0 # Eric M. Schmidt, Apr 01 2016

Definition edited by and more terms from Eric M. Schmidt, Apr 01 2016

A271210 Artiads (A001583) congruent to 1 mod 50 and for which 5 is a quintic residue.

Original entry on oeis.org

13451, 15901, 19001, 19801, 21701, 22901, 28001, 38851, 50551, 64301, 65101, 66851, 78101, 89101, 94351, 95701, 96401, 117751, 124001, 126001, 127951, 136601, 138401, 150301, 162251, 164701, 167051, 178301, 178501, 181001, 183301, 185051, 185401, 185951, 186301

Offset: 1

Eric M. Schmidt, Apr 02 2016

nonn

Hyperartiads (A270798) congruent to 1 mod 50.

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. 123-124, Theorem 3.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

Cf. A001583, A270798.

def isa(n) : return n % 50 == 1 and is_prime(n) and 5.powermod((n-1)//5, n) == 1 and fibonacci((n - 1)//5) % n == 0

A030430 Primes of the form 10*n+1.

Original entry on oeis.org

11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331, 401, 421, 431, 461, 491, 521, 541, 571, 601, 631, 641, 661, 691, 701, 751, 761, 811, 821, 881, 911, 941, 971, 991, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291

Offset: 1

Warut Roonguthai

Also primes of form 5*n+1 or equivalently 5*n+6.
Primes p such that the arithmetic mean of divisors of p^4 is an integer: A000203(p^4)/A000005(p^4) = C. - Ctibor O. Zizka, Sep 15 2008
Being a subset of A141158, this is also a subset of the primes of form x^2-5*y^2. - Tito Piezas III, Dec 28 2008
5 is quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Primes p such that 5 divides sigma(p^4), cf. A274397. - M. F. Hasler, Jul 10 2016

Michael B. Porter, Table of n, a(n) for n = 1..100000
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 519.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Cf. A024912, A045453, A049511, A081759, A017281, A010051, A004615 (multiplicative closure).
Cf. A001583 (subsequence).
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009

a030430 n = a030430_list !! (n-1)
a030430_list = filter ((== 1) . a010051) a017281_list
-- Reinhard Zumkeller, Apr 16 2012

Select[Prime@Range[210], Mod[ #, 10] == 1 &] (* Ray Chandler, Dec 06 2006 *)
Select[Range[11,1291,10],PrimeQ] (*Zak Seidov, Aug 14 2011*)

is(n)=n%10==1 && isprime(n) \\ Charles R Greathouse IV, Sep 06 2012

lista(nn) = forprime(p=11, nn, if(p%10==1, print1(p, ", "))) \\ Iain Fox, Dec 30 2017

a(n) = 10*A024912(n)+1 = 5*A081759(n)+6.
A104146(floor(a(n)/10)) = 1.
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009
a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012
Intersection of A000040 and A017281. - Iain Fox, Dec 30 2017

A047650 Primes for which golden mean tau is a quadratic residue.

Original entry on oeis.org

29, 89, 101, 181, 229, 349, 401, 461, 509, 521, 541, 709, 761, 769, 809, 941, 1009, 1021, 1049, 1061, 1109, 1229, 1249, 1289, 1361, 1409, 1549, 1601, 1621, 1669, 1709, 1721, 1741, 1789, 1861, 2029, 2069, 2081, 2089, 2389, 2441, 2621, 2729, 2801, 2861

Offset: 0

N. J. A. Sloane

Primes of the form x^2 + 20*y^2. - T. D. Noe, May 08 2005
Also primes p that divide the sum of cubes of the first (p-1)/2 Fibonacci numbers A005968((p-1)/2). - Alexander Adamchuk, Aug 07 2006
From A.H.M. Smeets, Nov 16 2023: (Start)
Mean gap size between two consecutive terms at p: ~ 8*log(p).
In x^2 + 20y^2: x == 1 (mod 2) and x !== 5 (mod 10). Otherwise not prime. (End)

A.H.M. Smeets, Table of n, a(n) for n = 0..20000 (terms 0..1700 from Vincenzo Librandi)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138.
E. Lehmer, Correction, Fib. Quart., 4 (1966), 135-138.
E. Lehmer, On the quadratic character of the Fibonacci root (annotated scanned copy)

Cf. A047651, A001583.

Cf. A005968.

k:=20; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Vincenzo Librandi, Sep 05 2016

nn=20; pMax=3000; Union[Reap[Do[p=x^2 + nn*y^2; If[p<=pMax&&PrimeQ[p], Sow[p]], {x, Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2, 1]]] (* Vincenzo Librandi, Sep 05 2016 *)

From A.H.M. Smeets, Nov 16 2023: (Start)
Equals {prime(n): A296240(n) in {2^k: k > 0}} = {A308787} union {A308789} union {A308793} union ... .
a(n) ~ A000040(8*n). (End)

More terms from James Sellers, Jan 25 2000

A270800 Septic artiads: primes p congruent to 1 mod 14 for which all solutions of the congruence x^3 + x^2 - 2x - 1 == 0 (mod p) are 7th power residues.

Original entry on oeis.org

14197, 21617, 23801, 24977, 25999, 34763, 37549, 41959, 42407, 45053, 45599, 54713, 55987, 56099, 60271, 61657, 63463, 66067, 72577, 75307, 76343, 76777, 79283, 83357, 88397, 90469, 91309, 99611, 107927, 111217, 111301, 111791, 124699, 126127, 131251, 132287

Offset: 1

N. J. A. Sloane, Apr 01 2016

nonn

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 page 126 (but beware errors).
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

Cf. A001583.

def is_septic_artiad(n) :
    if not (n % 14 == 1 and is_prime(n)) : return False
    R. = PolynomialRing(GF(n))
    return 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 added and sequence extended and corrected by Eric M. Schmidt, Apr 02 2016

A047652 Primes for which golden mean is a cubic residue.

Original entry on oeis.org

139, 151, 199, 331, 541, 619, 661, 709, 811, 829, 919, 1069, 1231, 1279, 1291, 1381, 1471, 1579, 1699, 1999, 2161, 2221, 2239, 2251, 2281, 2371, 2389, 2521, 2659, 2689, 2749, 3001, 3121, 3271, 3331, 3391, 3499, 3529, 3571, 3631, 3919, 4021, 4051, 4159

Offset: 1

N. J. A. Sloane

Primes of the form x^2 + xy + 34y^2, whose discriminant is -135. - T. D. Noe, May 17 2005

Primes of the form x^2 + 135*y^2. - Arkadiusz Wesolowski, May 31 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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.

Select[Prime[Range[1000]],IntegerQ[Fibonacci[(#1-1)/3]/#1]&] (* Alexander Adamchuk, Sep 16 2006 *)

Primes p that divide Fibonacci((p-1)/3). - Alexander Adamchuk, Sep 16 2006

More terms from James Sellers, Jan 25 2000

A125253 Primes p that divide Fibonacci[(p-1)/7].

Original entry on oeis.org

2269, 2521, 2731, 2969, 3571, 3739, 4481, 4831, 5741, 6091, 6329, 6581, 9521, 10949, 11159, 11789, 12391, 13049, 13679, 14281, 14449, 14771, 16829, 16871, 18229, 19489, 19559, 20021, 20399, 21701, 23269, 24179, 24571, 26111, 29191, 31039

Offset: 1

Alexander Adamchuk, Nov 26 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A125252 = Primes p that divide Fibonacci[(p+1)/7]. Cf. A122487 = Primes p that divide Fibonacci[(p+1)/2]. Cf. A047652 = Primes p that divide Fibonacci[(p-1)/3]. Cf. A001583 = Artiads. Primes p that divide Fibonacci[(p-1)/5].

Select[Prime[Range[5000]], IntegerQ[Fibonacci[(#1-1)/7]/#1]&]

A270798 Hyperartiads.

Original entry on oeis.org

5281, 5591, 6211, 6271, 8581, 8861, 9011, 9661, 10391, 10691, 11621, 12011, 12911, 13451, 15901, 19001, 19801, 20521, 20921, 21481, 21701, 22901, 22921, 23371, 26141, 27241, 27481, 28001, 28711, 29131, 30971, 31321, 31511, 32341, 32381, 34211, 38261, 38611

Offset: 1

N. J. A. Sloane, Mar 31 2016

Artiads (A001583) for which 5 is a quintic residue. [Lehmer] - Eric M. Schmidt, Apr 01 2016

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131. Beware errors!
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

Cf. A001583.

def is_hyperartiad(n) : return n % 10 == 1 and is_prime(n) and 5.powermod((n-1)//5, n) == 1 and fibonacci((n-1)//5) % n == 0 # Eric M. Schmidt, Apr 01 2016

Extended and corrected by Eric M. Schmidt, Apr 01 2016

A168171 Least prime p = 1 (mod n) which divides Fibonacci((p-1)/n).

Original entry on oeis.org

11, 29, 139, 61, 211, 541, 2269, 89, 199, 281, 859, 661, 911, 2269, 2221, 2081, 2789, 2161, 3041, 421, 2521, 19009, 21529, 3001, 9901, 5981, 2161, 2269, 26449, 2221, 31249, 19681, 17491, 2789, 3571, 25309, 30859, 3041, 6709, 3001, 9349, 2521, 13159, 19009

Offset: 1

M. F. Hasler, Nov 25 2009

nonn

			For n=1, all numbers p satisfy p=1 (mod n), but p=11 is the least prime that divides F((p-1)/1)=F(p-1)=F(10)=55.
For n=2, all odd numbers, thus all primes p>2, satisfy p=1 (mod n), but p=29 is the first one to divide F((p-1)/2) = F(14) = 377 = 13*29.
For n=5, a(n)=211 is the smallest Artiad, i.e. prime p=1 (mod 5) which divides F((p-1)/5) = F(42) = 211*1269736.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

a[1] = 11;
a[n_] := For[p = 1, True, p = p + n, If[PrimeQ[p] && Divisible[Fibonacci[(p - 1)/n], p], Return[p]]];
a /@ Range[100] (* Jean-François Alcover, Oct 14 2019 *)

for(n=1,99,forprime(p=1,oo,(p-1)%n & next; fibonacci((p-1)/n)%p || print1(p, ", ") || next(2)))

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

Original entry on oeis.org

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

cp's OEIS Frontend

A270799 Artiads (A001583) congruent to 1 mod 50 and having 2 as a quintic residue.

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A271210 Artiads (A001583) congruent to 1 mod 50 and for which 5 is a quintic residue.

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Sage

A030430 Primes of the form 10*n+1.

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PARI

PARI

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A047650 Primes for which golden mean tau is a quadratic residue.

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Magma

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A270800 Septic artiads: primes p congruent to 1 mod 14 for which all solutions of the congruence x^3 + x^2 - 2x - 1 == 0 (mod p) are 7th power residues.

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Sage

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A047652 Primes for which golden mean is a cubic residue.

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Mathematica

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A125253 Primes p that divide Fibonacci[(p-1)/7].

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Mathematica

A270798 Hyperartiads.

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