A003631 -id:A003631 - cp's OEIS frontend

A051834 Fibonacci(Pn-1) mod Pn, where Pn is the n-th prime.

1, 1, 3, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1

Offset: 1

Jud McCranie, Dec 11 1999

nonn

Terms are 0 when Pn == 1 or 4 mod 5, terms are 1 when Pn == 2 or 3 mod 5.

			P3=5, Fibonacci(5-1)=3 mod 5.

Cf. A000045, A003631, A045468, A051830, A051831.

A106284 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has no zeros.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 31, 37, 41, 53, 71, 79, 83, 107, 151, 157, 199, 229, 233, 239, 241, 257, 263, 277, 281, 311, 317, 331, 337, 379, 389, 409, 431, 433, 463, 467, 521, 523, 541, 547, 557, 563, 571, 577, 607, 631, 659, 677, 727, 769, 787, 809, 827, 839, 853

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5-step sequences, A001591 and A074048.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106278 (number of distinct zeros of x^5-x^4-x^3-x^2-x-1 mod prime(n)), A106298, A106304 (period of Lucas and Fibonacci 5-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible mod p).

P:= x^5-x^4-x^3-x^2-x-1:
select(p -> [msolve(P,p)] = [], [seq(ithprime(i),i=1..10000)]); # Robert Israel, Mar 13 2024

t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^5-x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 200}];Prime[Flatten[Position[t, 0]]]

from itertools import islice
from sympy import Poly, nextprime
from sympy.abc import x
def A106284_gen(): # generator of terms
    from sympy.abc import x
    p = 2
    while True:
        if len(Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=p).ground_roots())==0:
            yield p
        p = nextprime(p)
A106284_list = list(islice(A106284_gen(),20)) # Chai Wah Wu, Mar 14 2024

Name corrected by Robert Israel, Mar 13 2024

A128289 Composite terms in A128288(n) = A023163(n)/3 for n>1.

Original entry on oeis.org

1853, 9701, 10877, 17261, 23323, 27403, 75077, 80189, 113573, 120581, 161027, 162133, 163059, 196877, 213749, 291941, 361397, 400987, 427549, 482677, 635627, 667589, 941291, 1030373, 1033997, 1140701, 1196061, 1256293, 1751747, 1816363, 1842581, 2288453, 2662277

Offset: 1

Alexander Adamchuk, Feb 24 2007

nonn

3 divides A023163(n) for n>1. A023163(n) are the numbers n such that Fibonacci(n) == -2 (mod n).
Almost all terms of A128288 are prime that belong to A003631 = {2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97} Primes congruent to {2, 3} mod 5; that are also the primes p that divide Fibonacci(p+1).
a(3) = 10877 = 73*149 belongs to A069107 Composite n such that n divides Fibonacci(n+1).
a(3) = 10877 and a(4) = 17261 belong to A094395 Odd composite n such that n divides Fibonacci(n) + 1.

			a(1) = A128288(74) = 1853 = 17*109.
a(2) = 9701 = 89*109.
a(3) = 10877 = 73*149.
a(4) = 17261 = 41*421.
a(5) = 23323 = 83*281.

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

Cf. A128288, A002708, A023172, A023173, A023162, A023163 = numbers n such that Fib(n) == -2 (mod n). Cf. A003631, A069107, A094413, A094395 = Odd composite n such that n divides Fibonacci(n) + 1.

Do[ f = Mod[ Fibonacci[3n], 3n ]; If[ !PrimeQ[n] && f == 3n-2, Print[ {n, FactorInteger[n]} ]], {n,1,25000} ]

Two more terms from R. J. Mathar, Oct 08 2007

a(9)-a(33) from Amiram Eldar, Apr 07 2019

A177086 Semiprimes k that divide Fibonacci(k-1).

Original entry on oeis.org

1891, 4181, 8149, 13201, 15251, 17711, 40501, 51841, 64079, 64681, 67861, 68251, 78409, 88601, 88831, 90061, 96049, 97921, 115231, 118441, 145351, 146611, 153781, 191351, 197209, 218791, 219781, 254321, 272611, 302101, 303101

Offset: 1

Jonathan Vos Post, Dec 09 2010

This is the semiprime (A001358) analog of A045468. Now A045468 has a very simple characterization: it consists of the primes ending in 1 or 9. Can one say anything about the present sequence?

			46368/23 = 2016 = 2^5 * 3^2 * 7 so (24-1) | Fibonacci(24) but 24 is not semiprime, so is not in the sequence.
a(1) = 1891 = 31 * 61 is not in the sequence because 1891 divides Fibonacci(1891-1) = Fibonacci(1890).
a(21) = 146611 = 271 * 541 because 146611 | Fibonacci(146610).

Chai Wah Wu, Table of n, a(n) for n = 1..837

Cf. A000040, A000045, A001358, A069106, A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).

Cf. A177745 (semiprimes k that divide Fibonacci(k+1)).

Select[Range[310000],PrimeOmega[#]==2 && Divisible[Fibonacci[#-1],#]&] (* Harvey P. Dale, May 02 2016 *)

{k: k is in A001358 and k|A000045(k-1)} = A069106 INTERSECTION A001358.

A214888 Primes congruent to {2, 3} mod 11.

Original entry on oeis.org

2, 3, 13, 47, 79, 101, 113, 157, 167, 179, 211, 223, 233, 277, 311, 409, 421, 431, 443, 487, 509, 541, 563, 607, 619, 641, 673, 739, 751, 761, 773, 827, 839, 883, 937, 971, 1069, 1091, 1103, 1201, 1213, 1223, 1279, 1289, 1301, 1367, 1399, 1433, 1487, 1499

Offset: 1

Vincenzo Librandi, Aug 03 2012

nonn

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

Cf. A000040, A003631, A045343.

[p: p in PrimesUpTo(2000) | p mod 11 in [2, 3]];

Select[Prime[Range[500]],MemberQ[{2,3},Mod[#,11]]&]

A214889 Primes congruent to {2, 3} mod 13.

Original entry on oeis.org

2, 3, 29, 41, 67, 107, 197, 211, 223, 263, 353, 367, 379, 419, 431, 457, 509, 523, 587, 601, 613, 653, 691, 743, 757, 769, 809, 821, 887, 977, 991, 1069, 1237, 1277, 1289, 1303, 1367, 1381, 1433, 1459, 1471, 1511, 1523, 1549, 1601, 1627, 1667, 1693, 1783

Offset: 1

Vincenzo Librandi, Aug 03 2012

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

Cf. A000040, A003631, A045343.

[p: p in PrimesUpTo(3000) | p mod 13 in [2, 3]];

Select[Prime[Range[2000]],MemberQ[{2,3},Mod[#,13]]&]

A214890 Primes congruent to {2, 3} mod 17.

Original entry on oeis.org

2, 3, 19, 37, 53, 71, 139, 173, 223, 241, 257, 359, 461, 479, 547, 563, 631, 683, 733, 751, 853, 887, 937, 971, 1039, 1091, 1193, 1277, 1447, 1481, 1499, 1549, 1567, 1583, 1601, 1669, 1753, 1787, 1873, 1889, 1907, 2111, 2161, 2179, 2213, 2281, 2297, 2383

Offset: 1

Vincenzo Librandi, Aug 03 2012

nonn

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

Cf. A000040, A003631, A045343.

[p: p in PrimesUpTo(3000) | p mod 17 in [2, 3]];

Select[Prime[Range[3000]],MemberQ[{2,3},Mod[#,17]]&]
Select[Flatten[(#+{2,3})&/@(17*Range[0,150])],PrimeQ] (* Harvey P. Dale, Aug 22 2020 *)

A035227 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 45.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 2, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 2, 1, 1

Offset: 1

N. J. A. Sloane

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001622, A003631, A045468.

a[n_] := DivisorSum[n, KroneckerSymbol[45, #] &]; Array[a, 100] (* Amiram Eldar, Nov 20 2023 *)

my(m = 45); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(45, d)); \\ Amiram Eldar, Nov 20 2023

From Amiram Eldar, Nov 20 2023: (Start)
a(n) = Sum_{d|n} Kronecker(45, d).
Multiplicative with a(p^e) = 1 if Kronecker(45, p) = 0 (p = 3 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(45, p) = -1 (p is in A003631 \ {3}), and a(p^e) = e+1 if Kronecker(45, p) = 1 (p is in A045468).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 8*log(phi)/(3*sqrt(5)) = 0.573878587952..., where phi is the golden ratio (A001622) . (End)

A168483 Terms of A123239 which are primes in Z(sqrt(5)).

Original entry on oeis.org

2, 3, 13, 37, 67, 73, 83, 103, 107, 157, 193, 227, 277, 307, 313, 347, 367, 373, 397, 433, 443, 467, 523, 547, 563, 577, 587, 613, 673, 683, 733, 757, 787, 827, 853, 877, 883, 947, 967, 997, 1093, 1117, 1163, 1187, 1213, 1223, 1237, 1283, 1297

Offset: 1

A.K. Devaraj, Nov 26 2009

nonn

Cf. A003631, A123239, A168367.

MangammalQ[p_] := Block[{k = 3}, While[k > 2, k = Mod[3 k, p]]; k != 2];
A168483 = Select[Prime[Range[215]], MangammalQ[#] && MemberQ[{2, 3}, Mod[#, 5]] &] (* Ray Chandler, Jul 21 2011 *)

Corrected and extended by Ray Chandler, Jul 21 2011

A177745 Semiprimes k that divide Fibonacci(k+1).

Original entry on oeis.org

323, 377, 3827, 5777, 10877, 11663, 18407, 19043, 23407, 25877, 27323, 34943, 39203, 51983, 53663, 60377, 75077, 86063, 94667, 100127, 113573, 121103, 121393, 161027, 162133, 182513, 195227, 200147, 231703, 240239, 250277, 294527, 306287, 345913, 381923, 429263, 430127, 454607, 500207, 507527, 548627, 569087, 600767, 635627, 636707, 685583, 697883, 736163, 753377, 775207, 828827, 851927, 948433, 983903

Offset: 1

Jonathan Vos Post, Dec 12 2010

Data from T. D. Noe.

			a(1) = 323 = 17 * 19 because it is semiprime (product of two prime A000040), and 323 divides F(324) = 23041483585524168262220906489642018075101617466780496790573690289968, with dividend 2^4 * 3^5 * 53 * 107 * 109 * 2269 * 3079 * 4373 * 5779 * 19441 * 11128427 * 62650261 * 1828620361 * 6782976947987.

Cf. A177086, A000045, A001358, A069106, A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).

With[{semis=Select[Range[1000000],PrimeOmega[#]==2&]},Select[semis, Divisible[Fibonacci[#+1],#]&]] (* Harvey P. Dale, Aug 20 2012 *)

{k: k is in A001358 and k|A000045(k+1)}.

cp's OEIS Frontend

A051834 Fibonacci(Pn-1) mod Pn, where Pn is the n-th prime.

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A106284 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has no zeros.

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A128289 Composite terms in A128288(n) = A023163(n)/3 for n>1.

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A177086 Semiprimes k that divide Fibonacci(k-1).

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A214888 Primes congruent to {2, 3} mod 11.

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Magma

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A214889 Primes congruent to {2, 3} mod 13.

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Magma

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A214890 Primes congruent to {2, 3} mod 17.

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Magma

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A035227 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 45.

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A035227 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 45.