A003631 -id:A003631 - cp's OEIS frontend

A274397 Positive integers m such that sigma(m) is divisible by 5.

8, 19, 24, 27, 29, 38, 40, 54, 56, 57, 58, 59, 72, 76, 79, 87, 88, 89, 95, 104, 108, 109, 114, 116, 118, 120, 128, 133, 135, 136, 139, 145, 149, 152, 158, 168, 171, 174, 177, 178, 179, 184, 189, 190, 199, 200, 203, 209, 216, 218, 228, 229, 232, 236, 237, 239, 247, 248, 261, 264, 266, 267, 269, 270, 278, 280, 285, 290, 295, 296, 297

Offset: 1

M. F. Hasler, Jul 02 2016

nonn

See the subsequence A274685 of odd terms for a remark on frequent pairs of the form (30k-3, 30k-1).
If m is in the sequence and gcd(k,m)=1, then k*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The primitive terms are the primes and powers of primes within the sequence, cf. below.
Integers m > 0 where an integer k exists such that A000203(m) = A008587(k). - Felix Fröhlich, Jul 02 2016
For any prime p <> 5 there is an exponent k in {1, 3, 4} (depending on whether p is in A030433, A003631 or A030430) such that p^k is in this sequence. Given these p^k, the sequence consists of all numbers of the form n*p^(q*(k+1)-1) where n is coprime to p and q >= 1. Otherwise said, all numbers m which have some prime factor p with multiplicity q*(k+1)-1, where k = k(p) in {1, 3, 4} as introduced before. - M. F. Hasler, Jul 10 2016

			Some values for a(2^k): We have a(2) = 19, a(4) = 27, a(8) = 54, a(16) = 87, a(32) = 145, a(64) = 270, a(128) = 488, a(256) = 919, a(512) = 1736, a(1024) = 3267, a(2048) = 6258, a(4096) = 12035, a(8192) = 23160, a(16384) = 44878, a(32768) = 87207, a(65536) = 169911, a(131072) = 332009, a(262144) = 650031, a(524288) = 1274569, a(1048576) = 2503510, a(2097152) = 4924370, a(4194304) = 9697475, a(8388608) = 19116191.

Robert Israel, Table of n, a(n) for n = 1..10000
Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 20.
N. J. A. Sloane, Needed: smallest number k with sigma(sigma(k)) = 5k, SeqFan list, Jul 02 2016.

Cf. A000203, A028983 (sigma even), A087943 (sigma = 3k), A248150 (sigma = 4k); A028982 (sigma is odd), A248151 (sigma is not divisible by 4); A272930 (sigma(sigma(k)) = nk).

select(t -> numtheory:-sigma(t) mod 5 = 0, [$1..1000]); # Robert Israel, Jul 12 2016

Select[Range[300], Divisible[DivisorSigma[1, #], 5]&] (* Jean-François Alcover, Apr 09 2019 *)

PARI
```
is(n)=sigma(n)%5==0
```

is(n)=for(i=1,#n=factor(n)~,n[1,i] != 5 && (n[2,i]+1) % [5,4,4,2][n[1,i]%5] == 0 && return(1))

lim_{n->oo} a(k)/k = 2 (conjectured; cf. Examples).

Edited by M. F. Hasler, Jul 10 2016

A051831 a(n) = Fibonacci(prime(n)) mod prime(n), where prime(n) is the n-th prime.

Original entry on oeis.org

1, 2, 0, 6, 1, 12, 16, 1, 22, 1, 1, 36, 1, 42, 46, 52, 1, 1, 66, 1, 72, 1, 82, 1, 96, 1, 102, 106, 1, 112, 126, 1, 136, 1, 1, 1, 156, 162, 166, 172, 1, 1, 1, 192, 196, 1, 1, 222, 226, 1, 232, 1, 1, 1, 256, 262, 1, 1, 276, 1, 282, 292, 306, 1, 312, 316, 1, 336, 346, 1, 352, 1

Offset: 1

Jud McCranie, Dec 11 1999

nonn

Terms are 1 when prime(n) == 1 or 4 mod 5, terms are prime(n)-1 when prime(n) == 2 or 3 mod 5.

In general, it appears that Fibonacci(k*p) mod p = Fibonacci(k) or p-Fibonacci(k) for prime p > Fibonacci(k). For example Fibonacci(8*29) mod 29 = 21. - Gary Detlefs, May 28 2014

			prime(3) = 5, fibonacci(5) = 5 == 0 mod 5.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
R. Peele and P. Stanica, Matrix powers of column-justified Pascal triangles and Fibonacci sequences, arXiv:math/0010186 [math.CO], 2000.

Cf. A000045, A003631, A045468, A051834, A051830, A236395.

p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, ithprime(n)$2)[1, 2]:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 10 2015

Mod[Fibonacci[Prime[#]],Prime[#]]&/@Range[75] (* Harvey P. Dale, Jan 14 2011 *)

vector(80, n, fibonacci(prime(n)) % prime(n)) \\ Michel Marcus, Jul 15 2015

A182554 Composite numbers k that divide Fibonacci(k+1) or Fibonacci(k-1).

Original entry on oeis.org

323, 377, 442, 1891, 2737, 2834, 3827, 4181, 5777, 6479, 6601, 6721, 8149, 10877, 11663, 13201, 13981, 15251, 17119, 17711, 18407, 19043, 20999, 23407, 25877, 27323, 30889, 34561, 34943, 35207, 39203, 40501, 44099, 47519, 50183, 51841, 51983, 52701, 53663

Offset: 1

Gary Detlefs, May 04 2012

nonn

Pseudoprimes to a Fibonacci criterion for primality.
It is known that for prime p <> 5, Fibonacci(p-1) or Fibonacci(p+1) is divisible by p. (see Burton reference)
Primes for which Fibonacci(p-1) are divisible by p are congruent to {0,1,4} mod 5 and are listed in A038872.
Primes for which Fibonacci(p+1) are divisible by p are congruent to {2,3} mod 5 and are listed in A003631.
For n <= 1000, a(n) is squarefree (see A005117). - Dmitry Kamenetsky, Jul 20 2015
Any nonsquarefree term is divisible by the square of a Fibonacci-Wieferich prime (i.e., a prime p such that Fibonacci(j) == 0 (mod p^2) for some j not divisible by p). No Fibonacci-Wieferich primes are known, and there are none < 2*10^14, although it is conjectured that there are infinitely many. - Robert Israel, Jul 22 2015

David M. Burton, Elementary Number Theory, Allyn and Bacon, 1980, p. 292, #1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Giovanni Resta)
R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. 76 (2007), 2087-2094.

Cf. A038872, A003631, A000040, A094395, A005117.

with(combinat): f:= n-> fibonacci(n): for n from 1 to 40000 do if (f(n+1)/n = floor(f(n+1)/n) or f(n-1)/n=floor(f(n-1)/n)) and not isprime(n) then print(n) fi od;

Select[Range[50000], CompositeQ[#] && (Divisible[Fibonacci[# - 1], #] || Divisible[Fibonacci[# + 1], #]) &] (* Amiram Eldar, Sep 12 2022 *)

p=2;forprime(q=3,1e5,for(n=p+1,q-1,t=Mod([1,1;1,0],n)^(n-1);if(t[1,2]==0 || (t*[1,1;1,0]^2)[1,2]==0,print1(n", ")));p=q) \\ Charles R Greathouse IV, May 05 2012

A002382 Numbers of the form (p^2 - 49)/120 where p is prime.

Original entry on oeis.org

0, 1, 2, 4, 11, 15, 18, 23, 37, 44, 57, 78, 88, 95, 106, 134, 156, 205, 221, 232, 249, 310, 323, 414, 429, 452, 550, 576, 639, 667, 715, 785, 816, 837, 946, 1003, 1038, 1122, 1159, 1222, 1313, 1562, 1635, 1740, 1786, 1817, 1976, 2108, 2279, 2493, 2585, 2641

Offset: 1

N. J. A. Sloane

nonn

Primes p corresponding to a(n) are found in A003631(n+2) = A042993(n+3) = A097957(n+1). - Ray Chandler, Jul 29 2019

H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384.
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).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384. [Annotated scanned copy]

Cf. A002381, A002855, A003631, A042993, A097957.

Select[(Prime[Range[150]]^2-49)/120,IntegerQ] (* Harvey P. Dale, Jan 19 2014 *)

More terms from James Sellers, May 03 2000

A035202 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 20.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also number of divisors of n which end in 1 or 9 minus number of divisors of n which end in 3 or 7. E.g. a(98)=2-1=1 since divisors of 98 are: 1 and 49 counting +1 each; 2, 14 and 98 counting 0 each; and 7 counting -1. - Henry Bottomley, Jul 08 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000
MathNerds, An Excess of Divisors. [Wayback Machine link]

Cf. A083911, A083913, A083917, A083919.

Cf. A001622, A003631, A045468.

a:= proc(n) local D,d; D:= map(`modp`,convert(numtheory:-divisors(n),list),10);
      numboccur(1,D) + numboccur(9,D) - numboccur(3,D) - numboccur(7,D);
end proc:
seq(a(n),n=1..1000); # Robert Israel, Sep 22 2014

a[n_] := With[{d = Mod[Divisors[n], 10]}, Count[d, 1|9] - Count[d, 3|7]];
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 15 2023 *)
a[n_] := DivisorSum[n, KroneckerSymbol[20, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)

my(m = 20); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(20, d)); \\ Amiram Eldar, Nov 19 2023

From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(20, d).
Multiplicative with a(p^e) = 1 if Kronecker(20, p) = 0 (p = 2 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(20, p) = -1 (p is in A003631 \ {2}), and a(p^e) = e+1 if Kronecker(20, p) = 1 (p is in A045468).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*log(phi)/sqrt(5) = 0.645613411446..., where phi is the golden ratio (A001622). (End)

More terms from Henry Bottomley, Jul 08 2003

A051830 a(n) = Fibonacci(p(n)+1) mod p(n), where p(n) is the n-th prime.

Original entry on oeis.org

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

Offset: 1

Jud McCranie, Dec 11 1999

nonn

Terms are 1 when p(n) == 1 or 4 (mod 5) and 0 when p(n) == 2 or 3 (mod 5).

			p(3) = 5, so a(3) = Fibonacci(5+1) mod 5 = 3.

Cf. A000040, A000045, A003631, A045468, A051834, A051831.

Table[Mod[Fibonacci[n+1],n],{n,Prime[Range[110]]}] (* Harvey P. Dale, Nov 27 2015 *)

a(n) = max(0, Legendre(5,prime(n))) for n >= 4, where Legendre is the Legendre symbol. - Haifeng Xu, Jan 31 2025

A097957 Primes p such that p divides 5^((p-1)/2) + 4^((p-1)/2).

Original entry on oeis.org

3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 433, 443, 457, 463, 467, 487, 503, 523, 547, 557, 563, 577, 587

Offset: 1

Cino Hilliard, Sep 06 2004

Also odd primes congruent to {2, 3} mod 5, or primes with last digit 3 or 7. - Alexander Adamchuk, Nov 02 2006

5 is not a square mod p. - Michael Somos, Aug 15 2012

			5^3 + 4^3 = 7*27

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

Cf. A003631 = Primes congruent to {2, 3} mod 5.

Select[Prime[Range[120]],Divisible[5^((#-1)/2)+4^((#-1)/2),#]&] (* Harvey P. Dale, Feb 25 2013 *)

\s = +-1,d=diff ptopm1d2(n,x,d,s) = { forprime(p=3,n,p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0,print1(p","))) }

{a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( 20, m )== -1, cnt++ )); m} /* Michael Somos, Aug 15 2012 */

a(n) = A003631(n-1). - Alexander Adamchuk, Nov 02 2006

Definition clarified by Harvey P. Dale, Feb 25 2013

A100993 Indices k of Fibonacci numbers F(k) (A000045) that are divisible by k-1.

Original entry on oeis.org

2, 3, 4, 8, 14, 18, 24, 38, 44, 48, 54, 68, 74, 84, 98, 104, 108, 114, 128, 138, 158, 164, 168, 174, 194, 198, 224, 228, 234, 258, 264, 278, 284, 294, 308, 314, 318, 324, 338, 348, 354, 368, 374, 378, 384, 398, 434, 444, 458, 464, 468, 488, 504, 524, 548, 558

Offset: 1

Ron Knott, Nov 25 2004

nonn

When k-1 is prime, it is in A003631; when k-1 is composite (such as 323), it is in A069107. - T. D. Noe, Dec 13 2004

			14 is a term because F(14) = 377 = 13*29 is divisible by 13, one less than its index number 14.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ron Knott, Mathematical Magic of the Fibonacci Numbers.

Cf. A000045, A100992, A003631, A023162, A023173, A069104, A069107.

Select[Range[2, 500], Mod[ Fibonacci[ # ], # - 1] == 0 &] (* Robert G. Wilson v, Nov 26 2004 *)

a(n) = A069104(n) + 1. - T. D. Noe, Dec 13 2004

A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros.

Original entry on oeis.org

2, 5, 11, 13, 31, 43, 53, 79, 83, 89, 97, 103, 109, 131, 139, 151, 197, 199, 229, 233, 239, 251, 257, 271, 283, 313, 317, 347, 359, 367, 379, 389, 433, 443, 461, 479, 487, 521, 569, 571, 577, 593, 599, 601, 617, 631, 641, 643, 647, 659, 673, 677, 719, 769, 797

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817.

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

Cf. A106277 (number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n)), A106296 (period of Lucas 4-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible in mod p).

Res:= NULL: count:= 0: p:= 0:
P:= x^4 - x^3 - x^2 - x - 1:
while count < 100 do
  p:= nextprime(p);
  if [msolve(P,p)] = [] then
    Res:= Res, p; count:= count+1;
  fi
od:
Res; # Robert Israel, Mar 13 2024

t=Table[p=Prime[n]; cnt=0; Do[If[Mod[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 A106283_gen(): # generator of terms
    p = 2
    while True:
        if len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=p).ground_roots())==0:
            yield p
        p = nextprime(p)
A106283_list = list(islice(A106283_gen(),20)) # Chai Wah Wu, Mar 14 2024

Name corrected by Robert Israel, Mar 13 2024

A128288 a(n) = A023163(n)/3 for n > 1.

Original entry on oeis.org

3, 13, 37, 43, 53, 67, 83, 107, 157, 163, 173, 197, 227, 277, 283, 293, 307, 317, 347, 373, 397, 443, 467, 523, 547, 557, 563, 587, 613, 643, 653, 677, 683, 733, 757, 773, 787, 797, 827, 853, 877, 883, 907, 947, 997, 1013, 1093, 1117, 1123, 1163, 1187, 1213

Offset: 2

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 {a(n)} 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). The first composite term is a(74) = 1853 = 17*109. The second composite term is 9701 = 89*109. The third composite term is 10877 = 73*149 belong to A069107(n) Composite n such that n divides F(n+1) where F(k) are the Fibonacci numbers. Composite terms in {a(n)} are listed in A128289 = {1853, 9701, 10877, 17261, ...}.

			A023163 begins {1, 9, 39, 111, 129, 159, 201, 249, 321, 471, 489, 519, ...}.
Thus a(2) = A023163(2)/3 = 9/3 = 3, a(3) = A023163(3)/3 = 39/3 = 13.

Cf. A002708, A023172, A023173, A023162, A023163 (numbers k such that Fibonacci(k) == -2 (mod k)).

Cf. A003631, A069107, A128289 (composite terms in A128288).

a(n) = A023163(n)/3 for n > 1.

cp's OEIS Frontend

A274397 Positive integers m such that sigma(m) is divisible by 5.

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A051831 a(n) = Fibonacci(prime(n)) mod prime(n), where prime(n) is the n-th prime.

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A182554 Composite numbers k that divide Fibonacci(k+1) or Fibonacci(k-1).

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A002382 Numbers of the form (p^2 - 49)/120 where p is prime.

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A035202 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 20.

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A051830 a(n) = Fibonacci(p(n)+1) mod p(n), where p(n) is the n-th prime.

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A097957 Primes p such that p divides 5^((p-1)/2) + 4^((p-1)/2).

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A035202 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 20.