A053027 -id:A053027 - cp's OEIS frontend

A122870 Primes congruent to 3 or 7 mod 20.

3, 7, 23, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 263, 283, 307, 347, 367, 383, 443, 463, 467, 487, 503, 523, 547, 563, 587, 607, 643, 647, 683, 727, 743, 787, 823, 827, 863, 883, 887, 907, 947, 967, 983, 1063, 1087, 1103, 1123, 1163, 1187, 1223

Offset: 1

Alexander Adamchuk, Sep 16 2006

nonn

The old name was "Primes p that divide Lucas((p+1)/2) = A000032((p+1)/2)".

Note that F(p+1) = F((p+1)/2)*Lucas((p+1)/2), where F = A000045. Since gcd(F(n),Lucas(n)) = 1 or 2 (because Lucas(n)^2 - 5*F(n)^2 = 4*(-1)^n), this sequence (under the old definition above) lists primes p such that p divides F(p+1) but does not divides F((p+1)/2). By Propositions 1.1 and 1.2 (the k = 3 case) of my link below, this is primes p == 3, 7 (mod 20). - Jianing Song, Jun 20 2025

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jianing Song, Lucas sequences and entry point modulo p
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Gaussian Prime.

Cf. A000032, A000045, A122869, A216815.

Subseqeunce of A002145, A003631, A049098, A053027. Essentially the same as A106865.

[p: p in PrimesUpTo(1500) | p mod 20 in [3, 7]]; // Vincenzo Librandi, Jan 06 2013

Select[Prime[Range[1000]],IntegerQ[(Fibonacci[(#1+1)/2-1]+Fibonacci[(#1+1)/2+1])/#1]&]
Select[Prime[Range[300]], MemberQ[{3, 7}, Mod[#, 20]]&] (* Vincenzo Librandi, Jan 06 2013 *)

I merged A216816 into this entry at the suggestion of Jianing Song, Jun 20 2025. - N. J. A. Sloane, Jun 22 2025

A086598 Number of distinct prime factors in Lucas(n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 3, 1, 1, 3, 1, 2, 3, 3, 2, 3, 3, 2, 3, 2, 2, 4, 1, 2, 3, 3, 4, 4, 1, 2, 4, 3, 1, 5, 2, 4, 6, 3, 1, 4, 2, 4, 4, 3, 1, 4, 4, 2, 4, 3, 3, 6, 1, 2, 6, 2, 5, 5, 2, 2, 5, 4, 1, 4, 2, 3, 7, 2, 4, 4, 1, 2, 5, 4, 2, 6, 4, 2, 5, 3, 2, 6, 3, 3, 4, 4, 5, 4, 2, 4, 7, 4, 3, 6, 3, 4, 9

Offset: 0

T. D. Noe, Jul 24 2003

Interestingly, the Lucas numbers separate the primes into three disjoint sets: (A053028) primes that do not divide any Lucas number, (A053027) primes that divide Lucas numbers of even index and (A053032) primes that divide Lucas numbers of odd index.

Max Alekseyev, Table of n, a(n) for n = 0..1411 (first 1000 terms from T. D. Noe)
Blair Kelly, Fibonacci and Lucas Factorizations
Eric Weisstein's World of Mathematics, Lucas Number

Cf. A000204 (Lucas numbers), A086599 (number of prime factors, counting multiplicity), A086600 (number of primitive prime factors).

Cf. A053027, A053028, A053032.

[#PrimeDivisors(Lucas(n)): n in [1..100]]; // Vincenzo Librandi, Jul 26 2017

Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; Table[Length[FactorInteger[Lucas[n]]], {n, 150}]

a(n)=omega(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = Sum{d|n and n/d odd} A086600(d) + 1 if 6|n, a Mobius-like transform

a(0)=1 prepended by Max Alekseyev, Jun 15 2025

A140409 Prime factors of Lucas numbers.

Original entry on oeis.org

2, 3, 7, 11, 19, 23, 29, 31, 41, 43, 47, 59, 67, 71, 79, 83, 101, 103, 107, 127, 131, 139, 151, 163, 167, 179, 181, 191, 199, 211, 223, 227, 229, 239, 241, 251, 263, 271, 281, 283, 307, 311, 331, 347, 349, 359, 367, 379, 383, 401, 409, 419, 431, 439, 443, 449

Offset: 1

Tanya Khovanova, Jun 16 2008

nonn

a(n) is A096362(n) in ascending order.

C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Cf. A096362 Order in which prime factors first occur in the Lucas sequence.

Union of 2, A053027 and A053032 - T. D. Noe, Jun 21 2008

More terms from T. D. Noe, Jun 21 2008

A215850 Primes p such that 2*p + 1 divides Lucas(p).

Original entry on oeis.org

5, 29, 89, 179, 239, 359, 419, 509, 659, 719, 809, 1019, 1049, 1229, 1289, 1409, 1439, 1499, 1559, 1889, 2039, 2069, 2129, 2339, 2399, 2459, 2549, 2699, 2819, 2939, 2969, 3299, 3329, 3359, 3389, 3449, 3539, 3779, 4019, 4349, 4409, 4919, 5039, 5279, 5399, 5639

Offset: 1

Arkadiusz Wesolowski, Aug 24 2012

nonn

An equivalent definition of this sequence: 5 together with primes p such that p == -1 (mod 30) and 2*p + 1 is also prime.
Sequence without the initial 5 is the intersection of A005384 and A132236.
These numbers do not occur in A137715.
From Arkadiusz Wesolowski, Aug 25 2012: (Start)
The sequence contains numbers like 1409 which are in A053027.
a(n) is in A002515 if and only if a(n) is congruent to -1 mod 60. (End)

			29 is in the sequence since it is prime and 59 is a factor of Lucas(29) = 1149851.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
C. K. Caldwell, "Top Twenty" page, Lucas cofactor
Eric Weisstein's World of Mathematics, Lucas Number

Supersequence of A230809. Cf. A000032, A132236.

[5] cat [n: n in [29..5639 by 30] | IsPrime(n) and IsPrime(2*n+1)];

Select[Prime@Range[740], Divisible[LucasL[#], 2*# + 1] &]
Prepend[Select[Range[29, 5639, 30], PrimeQ[#] && PrimeQ[2*# + 1] &], 5]

is_A215850(n)=isprime(n)&!real((Mod(2,2*n+1)+quadgen(5))*quadgen(5)^n) \\ - M. F. Hasler, Aug 25 2012

A116515 a(n) = the period of the Fibonacci numbers modulo p divided by the smallest m such that p divides Fibonacci(m), where p is the n-th prime.

Original entry on oeis.org

1, 2, 4, 2, 1, 4, 4, 1, 2, 1, 1, 4, 2, 2, 2, 4, 1, 4, 2, 1, 4, 1, 2, 4, 4, 1, 2, 2, 4, 4, 2, 1, 4, 1, 4, 1, 4, 2, 2, 4, 1, 1, 1, 4, 4, 1, 1, 2, 2, 1, 4, 1, 2, 1, 4, 2, 4, 1, 4, 2, 2, 4, 2, 1, 4, 4, 1, 4, 2, 1, 4, 1, 2, 4, 1, 2, 4, 4, 2, 2, 1, 4, 1, 4, 1, 2, 2, 4, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 2, 2, 1

Offset: 1

Nick Krempel, Mar 24 2006

Conditions on p_n mod 4 and mod 5 restrict possible values of a(n). The unknown (?) case is p = 1 mod 4 and (5|p) = 1, equivalently, p = 1 or 9 mod 20, where {1, 2, 4} all occur.

Number of zeros in fundamental period of Fibonacci numbers mod prime(n). [From T. D. Noe, Jan 14 2009]

			a(4) = 2, as 7 is the 4th prime, the Fibonacci numbers mod 7 have period 16, the first Fibonacci number divisible by 7 is F(8) = 21 = 3*7 and 16 / 8 = 2.
One period of the Fibonacci numbers mod 7 is 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, which has two zeros. Hence a(4)=2. [From _T. D. Noe_, Jan 14 2009]

Cf. A060305, A001602.

Cf. A112860, A053027, A053028 (primes producing 1, 2 and 4 zeros) [From T. D. Noe, Jan 14 2009]

a(n) = A060305(n) / A001602(n). a(n) is always one of {1, 2, 4}.

a(n) = A001176(prime(n)) [From T. D. Noe, Jan 14 2009]

cp's OEIS Frontend

A122870 Primes congruent to 3 or 7 mod 20.

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A086598 Number of distinct prime factors in Lucas(n).

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A140409 Prime factors of Lucas numbers.

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A215850 Primes p such that 2*p + 1 divides Lucas(p).

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Magma

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A116515 a(n) = the period of the Fibonacci numbers modulo p divided by the smallest m such that p divides Fibonacci(m), where p is the n-th prime.

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