A001602 -id:A001602 - cp's OEIS frontend

A001603 Odd-indexed terms of A124296.

1, 11, 101, 781, 5611, 39161, 270281, 1857451, 12744061, 87382901, 599019851, 4105974961, 28143378001, 192899171531, 1322154751061, 9062194370461, 62113232767531, 425730505493801, 2918000490238361, 20000273409331051, 137083914639998701, 939587132382262661

Offset: 0

N. J. A. Sloane

Old name: Related to factors of Fibonacci numbers.

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).

John Cerkan, Table of n, a(n) for n = 0..1187
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 20.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (11, -33, 33, -11, 1).

Cf. A001604, A124296, A124297.

A001603:=-(1+13*z**2+z**4)/(z-1)/(z**2-3*z+1)/(z**2-7*z+1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

5 #^2 - 5 # + 1 &@ Fibonacci@ # & /@ Range[1, 43, 2] (* Michael De Vlieger, Apr 03 2017 *)

G.f.: -(1+13*x^2+x^4)/((x-1)*(x^2-3*x+1)*(x^2-7*x+1)). [After Simon Plouffe]

Entry revised by Michel Marcus and N. J. A. Sloane, Jun 06 2015

A069180 F(n) and n! are relatively prime where F(n) are the Fibonacci numbers.

Original entry on oeis.org

1, 2, 7, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 163, 166, 167, 169, 173, 178, 179, 181, 191

Offset: 1

Benoit Cloitre, Apr 10 2002

Are there any primes p >5 such that F(p) and p! are not relatively primes?
From Robert Israel, May 31 2018: (Start)
n is in the sequence if and only if there is no prime q = prime(k) <= n such that A001602(k) | n.
All primes > 5 are in the sequence, because A001602(k) < prime(k) for k > 3, and we can't have n prime unless A001602(k)=n.
(End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A001602.

N:= 200: # for all terms <= N
V:= Vector(N,1):
F:= proc(n) option remember; procname(n-1)+procname(n-2) end proc:
F(0):= 0: F(1):= 1:
K:= proc(q) local k;
   for k from 1 do if F(k) mod q = 0 then return k fi
     od
end proc:
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  k:= K(p);
  k0:= k*ceil(p/k);
  V[[seq(i,i=k0..N,k)]]:= 0
od:
select(t -> V[t]=1, [$1..N]); # Robert Israel, May 31 2018

Select[Range[1000], CoprimeQ[Fibonacci[#], #!]&] (* Jean-François Alcover, Jun 07 2020 *)

Conjecture : a(n) = C*n*Log(n) + 0(n*Log(n)) with 0, 6 < C < 0, 7

A073624 Primes ordered so that their most significant digits form the digits of Euler-Mascheroni constant .5772156649...

Original entry on oeis.org

5, 7, 71, 2, 11, 53, 61, 67, 41, 907, 13, 59, 3, 23, 83, 601, 607, 613, 503, 17, 2003, 9001, 89, 29, 401, 211, 43, 31, 101, 47, 223, 19, 509, 97, 37, 307, 521, 911, 311, 919, 929, 227, 313, 523, 937, 809, 8009, 541, 73, 617, 79, 229, 317, 409, 811, 821, 419, 823, 619

Offset: 1

Amarnath Murthy, Aug 08 2002

In case of a 0 use two digits.

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A001602, A073622, A073623.

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 25 2003

A194363 Lucas entry points: smallest m >= 0 such that the n-th prime divides Lucas(m), or -1 if there is no such m.

Original entry on oeis.org

0, 2, -1, 4, 5, -1, -1, 9, 12, 7, 15, -1, 10, 22, 8, -1, 29, -1, 34, 35, -1, 39, 42, -1, -1, 25, 52, 18, -1, -1, 64, 65, -1, 23, -1, 25, -1, 82, 84, -1, 89, 45, 95, -1, -1, 11, 21, 112, 114, 57, -1, 119, 60, 125, -1, 44, -1, 135, -1, 14, 142, -1, 22, 155, -1

Offset: 1

T. D. Noe, Oct 09 2011

sign

The -1 terms are for the primes in A053028. Note that 2 divides the zeroth Lucas number. In the plots, the uppermost line consists of the odd primes in A000057. Note that when a(n) > 0, then a(n) = A001602(n)/2.

T. D. Noe, Table of n, a(n) for n = 1..2000
T. D. Noe, Another plot of this sequence

Cf. A000204 (Lucas numbers), A001602 (Fibonacci entry points), A223486 (Lucas entry points), A000040 (prime numbers).

lim = 100; luc = LucasL[Range[0, Prime[lim]]]; Table[s = Select[Range[p], Mod[luc[[#]], p] == 0 &, 1]; If[s == {}, -1, s[[1]] - 1], {p, Prime[Range[lim]]}]

a(n) = A223486(A000040(n)). - Jon Maiga, Jul 01 2021

A230359 Prime numbers p such that their Fibonacci entry points are less than p+1.

Original entry on oeis.org

5, 11, 13, 17, 19, 29, 31, 37, 41, 47, 53, 59, 61, 71, 73, 79, 89, 97, 101, 107, 109, 113, 131, 137, 139, 149, 151, 157, 173, 179, 181, 191, 193, 197, 199, 211, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 373, 379, 389, 397, 401, 409, 419, 421, 431, 433, 439, 449, 457, 461, 479, 491, 499

Offset: 1

Brandon Avila and Tanya Khovanova, Oct 16 2013

nonn

For these primes p there exists a Fibonacci like sequence that doesn't contain multiples of p.

For other primes p the Fibonacci entry points are p+1. These primes are sequence A000057: Primes dividing all Fibonacci sequences.

Robert Israel, Table of n, a(n) for n = 1..10000
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.8.5.

A002144 is a subsequence.

Cf. A000057, A001177, A001602.

filter:= proc(n) local i,a,b,c;
  if not isprime(n) then return false fi;
  a:= 0; b:= 1;
  for i from 1 to n-1 do
   c:= b;
   b:= a+b mod n; if b = 0 then return true fi;
   a:= c;
  od;
false
end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Sep 01 2020

A001177[n_] := For[k = 1, True, k++, If[Divisible[Fibonacci[k], n], Return[k]]]; A230359 = Reap[For[p = 2, p <= 499, p = NextPrime[p], If[A001177[p] < 1+p, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 21 2013 *)

def isA230359(p):
    return any(p.divides(fibonacci(k)) for k in (1..p))
print([p for p in primes(1, 500) if isA230359(p)]) # Peter Luschny, Nov 01 2019

{p in A000040: A001177(p) < 1+p}.

A065106 Smallest Fibonacci index to produce a factor p^2 (for primes p).

Original entry on oeis.org

6, 12, 25, 56, 91, 110, 153, 342, 406, 552, 703, 752, 820, 915, 930, 979

Offset: 1

Len Smiley, Nov 21 2001

nonn

Following Lucas, these might be called the prime-squared ranks of apparition.

Assuming that there are no square primitive factors in the Fibonacci sequence (an open question), then this sequence continues 1431, 1892, 2147, 2701, 2943, 3029, 3422, 3852, 4378, 4556, 4753, 4970, 5050, 5513, 6162, 6394, 6972, 7550, 7868, 8841, 8862, 9453. This is obtained by sorting the sequence prime(n)*A001602(n). - T. D. Noe, Apr 15 2004

			342 is here but not in A065069 because Fib(342) is the first Fib divisible by 19^2, but 342 is divisible by 6 and so is not a primitive index.

Cf. A065069, A037917, A065107.

Cf. A001602 (smallest m such that prime(n) divides Fibonacci(m)).

A120947 a(n) = smallest m such that n-th prime divides Pell(m).

Original entry on oeis.org

2, 4, 3, 6, 12, 7, 8, 20, 22, 5, 30, 19, 10, 44, 46, 27, 20, 31, 68, 70, 36, 26, 84, 44, 48, 51, 34, 108, 55, 28, 126, 132, 17, 140, 75, 150, 79, 164, 166, 87, 36, 91, 190, 96, 9, 18, 212, 74, 76, 23, 116, 14, 40, 84, 64, 262, 15, 270, 139, 140, 284, 49, 308, 310, 78, 159, 332

Offset: 1

Ralf Stephan, Aug 19 2006

nonn

For all divisors d of n>0, Pell(d) divides Pell(n), so if a prime divides the n-th Pell number, so does it for all multiples of n.

For n > 1, a(n) is the multiplicative order of -3-2*sqrt(2), in GF(prime(n)) if 2 is a quadratic residue (mod prime(n)) or GF(prime(n)^2) otherwise. Thus a(n) divides prime(n)-1 if prime(n) == 1 or 7 (mod 8), i.e. n is in A024704, and a(n) divides prime(n)+1 if prime(n) == 3 or 5 (mod 8), i.e. n is 2 or is in A024705. - Robert Israel, Aug 28 2015

			a(4)=6 because the 6th Pell number, 70, is the first that is divisible by the 4th prime (=7).

Alois P. Heinz and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1 .. 1000 from Alois P. Heinz)
J. L. Schiffman, Exploring the Fibonacci sequence of order two with CAS technology, Paper C027, Electronic Proceedings of the Twenty-fourth Annual International Conference on Technology in Collegiate Mathematics, Orlando, Florida, March 22-25, 2012.

Cf. A000129 (Pell numbers), A001602 (equivalent sequence with Fibonacci numbers), A239111, A024704, A024705.

p:= proc(n) p(n):=`if`(n<2, n, 2*p(n-1)+p(n-2)) end:
a:= proc(n) local k, t; t:= ithprime(n);
      for k while irem(p(k), t)>0 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 28 2014
f:= proc(n)
local p, r, G;
uses numtheory;
p:= ithprime(n);
if quadres(2,p)=1 then
   r:= msqrt(2,p);
   order(-3-2*r, p)
else
   G:= GF(p, 2, r^2-2);
   G:-order( G:-ConvertIn(-3-2*r));
fi
end proc:
2, seq(f(n), n=2..100); # Robert Israel, Aug 28 2015

p[n_] := p[n] = If[n<2, n, 2*p[n-1] + p[n-2]]; a[n_] := Module[{k, t}, t = Prime[n]; For[k=1, Mod[p[k], t]>0, k++]; k]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 16 2015, after Alois P. Heinz *)

a(n,p=prime(n))=my(cur=Mod(1,p),last,m=1); while(cur, m++; [last,cur]=[cur,2*cur+last]); m \\ Charles R Greathouse IV, Jun 16 2015

A279011 Numbers k such that phi(6k) is either phi(6k-2) or phi(6k+2), where phi is Euler's totient function A000010.

Original entry on oeis.org

1, 2, 12, 152, 222, 268, 362, 432, 723, 992, 1517, 2532, 2567, 8472, 9718, 9858, 13498, 15738, 34732, 35898, 44092, 60363, 69312, 75168, 75973, 82752, 87208, 88888, 98198, 105852, 114392, 126848, 128672, 135368, 141093, 161268, 221223, 233788, 301513, 328358

Offset: 1

N. J. A. Sloane, Dec 10 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1554
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 67.

Cf. A000010.

Union of A279183 and A279184.

[n: n in [1..1000000] | not (EulerPhi(6*n) eq EulerPhi(6*n-2)) eq (EulerPhi(6*n) eq EulerPhi(6*n+2))]; // Vincenzo Librandi, Dec 12 2016

Select[Range[10^6], Function[k, Or @@ Map[EulerPhi[6 k] == EulerPhi@ # &, 6 k + {-2, 2}]]] (* Michael De Vlieger, Dec 12 2016 *)
Select[Range[330000],EulerPhi[6#]==EulerPhi[6#-2]||EulerPhi[6#]==EulerPhi[6#+2]&] (* Harvey P. Dale, Jul 07 2025 *)

More terms from Vincenzo Librandi, Dec 12 2016

A279183 Numbers k such that phi(6k) = phi(6k-2), where phi is Euler's totient function A000010.

Original entry on oeis.org

1, 2, 12, 152, 222, 362, 432, 992, 1517, 2532, 2567, 8472, 34732, 44092, 69312, 82752, 105852, 114392, 128672, 336992, 350082, 393132, 393552, 462747, 497712, 559872, 665817, 714502, 931432, 968952, 1126602, 1281867, 1389337, 1449992, 1638712, 1694292

Offset: 1

N. J. A. Sloane, Dec 10 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 1..761
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 67.

Cf. A000010.

A279011 is the union of A279183 and A279184.

[n: n in [1..2*10^6] | EulerPhi(6*n) eq EulerPhi(6*n-2)]; // Vincenzo Librandi, Dec 11 2016

a = {}; Do[If[EulerPhi[6k] == EulerPhi[6 k - 2], AppendTo[a, k]], {k, 1000000}]; a (* Vincenzo Librandi, Dec 11 2016 *)

isok(k) = eulerphi(6*k) == eulerphi(6*k-2); \\ Michel Marcus, Dec 11 2016

More terms from Vincenzo Librandi, Dec 11 2016

A279184 Numbers k such that phi(6k) = phi(6k+2), where phi is Euler's totient function A000010.

Original entry on oeis.org

268, 723, 9718, 9858, 13498, 15738, 35898, 60363, 75168, 75973, 87208, 88888, 98198, 126848, 135368, 141093, 161268, 221223, 233788, 301513, 328358, 330633, 419148, 507648, 527928, 543468, 551238, 556418, 586018, 725958, 772508, 964588, 985728

Offset: 1

N. J. A. Sloane, Dec 10 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 1..794
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 67.

Cf. A000010.

A279011 is the union of A279183 and A279184.

[n: n in [1..2*10^6] | EulerPhi(6*n) eq EulerPhi(6*n+2)]; // Vincenzo Librandi, Dec 11 2016

select( k -> numtheory:-phi(6*k)=numtheory:-phi(6*k+2), [$1..10^6]); # Robert Israel, Dec 11 2016

a = {}; Do[If[EulerPhi[6 k] == EulerPhi[6 k + 2], AppendTo[a, k]], {k, 1000000}]; a (* Vincenzo Librandi, Dec 11 2016 *)

isok(k) = eulerphi(6*k) == eulerphi(6*k+2); \\ Michel Marcus, Dec 11 2016

a(8)-a(33) from Robert Israel, Dec 11 2016

cp's OEIS Frontend

A001603 Odd-indexed terms of A124296.

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A069180 F(n) and n! are relatively prime where F(n) are the Fibonacci numbers.

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A073624 Primes ordered so that their most significant digits form the digits of Euler-Mascheroni constant .5772156649...

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A194363 Lucas entry points: smallest m >= 0 such that the n-th prime divides Lucas(m), or -1 if there is no such m.

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A230359 Prime numbers p such that their Fibonacci entry points are less than p+1.

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A065106 Smallest Fibonacci index to produce a factor p^2 (for primes p).

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A120947 a(n) = smallest m such that n-th prime divides Pell(m).

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A279011 Numbers k such that phi(6k) is either phi(6k-2) or phi(6k+2), where phi is Euler's totient function A000010.

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