A180363 -id:A180363 - cp's OEIS frontend

A285992 Primes in the bisected Lucas sequence A002878.

11, 29, 199, 521, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771, 258899611203303418721656157249445530046830073044201152332257717521

Offset: 1

R. J. Mathar, Apr 30 2017

nonn

Subsequence of A005479.

Cf. A001606, A002878, A005479, A121534, A180363, A269254, A294099 (row 3).

select(isprime, [seq(combinat:-fibonacci(2*n)+combinat:-fibonacci(2*n+2), n=1..200)]); # Robert Israel, May 01 2017

Select[LucasL[Range[1, 400, 2]], PrimeQ] (* Vincenzo Librandi, May 01 2017 *)
Select[LinearRecurrence[{3,-1},{1,4},160],PrimeQ] (* Harvey P. Dale, Sep 01 2024 *)

A002878 INTERSECT A000040.

A000032(k) for odd k in A001606. - Robert Israel, May 01 2017

A215809 Prime numbers n for which the Lucas number L(n) (see A000032) is the sum of two squares.

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 127, 163, 199, 223, 307, 313, 349, 397, 433, 523, 541, 613, 619, 709, 823, 907, 1087, 1123, 1129, 1213, 1279, 1531

Offset: 1

V. Raman, Aug 23 2012

nonn

These Lucas numbers L(n) have no prime factor congruent to 3 mod 4 to an odd power.
Also prime numbers n such that the Lucas number L(n) can be written in the form a^2 + 5*b^2.
Any prime factor of Lucas(n) for n prime is always of the form 1 (mod 10) or 9 (mod 10).
A number n can be written in the form a^2+5*b^2 (see A020669) if and only if n is 0,
or of the form 2^(2i) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m)
or of the form 2^(2i+1) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m+1),
for integers i,j,k,m, for primes p,q.
1607 <= a(34) <= 1747. 1747, 1951, 2053, 2467, 5107, 5419, 5851, 7243, 7741, 8467, 13963, 14449, 14887, 15511, 15907, 35449, 51169, 193201, 344293, 387433, 574219, 901657, 1051849 are terms. - Chai Wah Wu, Jul 22 2020

			Lucas(19) = 9349 = 95^2 + 18^2.
Lucas(19) = 9349 = 23^2 + 5*42^2.

Blair Kelly, Fibonacci and Lucas factorizations
Eric W. Weisstein, Sum of squares function

Cf. A000032, A001481, A124130, A180363, A215906, A215907.

Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).

forprime(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; has=0; for(j=1, #a, if(a[1, j]%4==3&&a[2, j]%2==1, has=1; break)); if(has==0, print(i", "))) \\ a^2+b^2 form.

forprime(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i", "))) \\ a^2+5*b^2 form.

Merged A215941 into this sequence, T. D. Noe, Sep 21 2012

a(30)-a(33) from Chai Wah Wu, Jul 22 2020

A215907 Odd numbers n such that the Lucas number L(n) is the sum of two squares.

Original entry on oeis.org

1, 3, 7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 111, 127, 163, 169, 183, 199, 223, 307, 313, 349, 361, 397, 433, 511, 523, 541, 613, 619, 709, 823, 907, 1087, 1123, 1129, 1147, 1213, 1279, 1434

Offset: 1

V. Raman, Aug 26 2012

These Lucas numbers L(n) have no prime factor congruent to 3 mod 4 to an odd power.
Also, numbers n such that L(n) can be written in the form a^2 + 5*b^2.
Subsequence of A124132.
Is this A124132 without the 6? - Joerg Arndt, Sep 07 2012
Any prime factor of Lucas(n) for the prime values of n is always of the form 1 (mod 10) or 9 (mod 10).
A number n can be written in the form a^2 + 5*b^2 if and only if n is 0, or of the form 2^(2i) 5^j Product_{p==1 or 9 mod 20} p^k Product_{q==3 or 7 mod 20) q^(2m) or of the form 2^(2i+1) 5^j Product_{p==1 or 9 mod 20} p^k Product_{q==3 or 7 mod 20) q^(2m+1), for integers i,j,k,m, for primes p,q.
1501 <= a(42) <= 1531. 1531, 1651, 1747, 1849, 1951, 2053, 2413, 2449, 2467, 4069, 5107, 5419, 5851, 7243, 7741, 8467, 13963, 14449, 14887, 15511, 15907, 35449, 51169, 193201, 344293, 387433, 574219, 901657, 1051849 are terms. - Chai Wah Wu, Jul 22 2020

			Lucas(19) = 9349 = 95^2 + 18^2.
Lucas(19) = 9349 = 23^2 + 5*42^2.

Blair Kelly, Fibonacci and Lucas factorizations
Eric W. Weisstein, Sum of squares function

Cf. A000032, A124130, A215809, A215906.
Cf. A180363.
Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).

for(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; has=0; for(j=1, #a, if(a[1, j]%4==3&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==1, print(i", "))) \\ a^2 + b^2 form.

for(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i", "))) \\ a^2 + 5*b^2 form.

17 more terms from V. Raman, Aug 28 2012
A215940 merged into this sequence by T. D. Noe, Sep 21 2012
a(38)-a(41) from Chai Wah Wu, Jul 22 2020

A268478 L(p) modulo p^2, where p = prime(n) and L is a Lucas number (A000032).

Original entry on oeis.org

3, 4, 11, 29, 78, 14, 103, 324, 70, 204, 497, 519, 1477, 1420, 1881, 902, 1476, 3600, 3418, 2202, 5257, 317, 914, 5074, 4269, 9192, 5666, 6421, 7086, 4182, 12193, 3800, 1097, 11677, 299, 22651, 17271, 12063, 18371, 26297, 13784, 10137, 8405, 33583, 11230

Offset: 1

Felix Fröhlich, Feb 05 2016

nonn

Lemma 7 from the Andrejic paper (p. 42): Prime p is a Wall-Sun-Sun prime iff L(p) == 1 (mod p^2). Therefore, a(n) = 1 iff A113650(n) = 0.

Felix Fröhlich, Table of n, a(n) for n = 1..10000
V. Andrejic, On Fibonacci powers, Publikacije Elektrotehnickog fakulteta - serija: matematika, 17 (2006), 38-44.

Cf. A000032, A000040, A113650, A180363.

[Lucas(p) mod p^2: p in PrimesUpTo(250)]; // Bruno Berselli, Feb 09 2016

Table[Mod[LucasL[Prime[n]], Prime[n]^2], {n, 60}] (* Vincenzo Librandi, Feb 09 2016 *)

a000032(n) = fibonacci(n+1) + fibonacci(n-1)
a(n) = my(p=prime(n)); lift(Mod(a000032(p), p^2))

a(n) = A180363(n) mod A001248(n). - Michel Marcus, Feb 09 2016

A160909 Row sums of triangle defined in A096539.

Original entry on oeis.org

2, 3, 10, 28, 198, 520, 3570, 9348, 64078, 1149850, 3010348, 54018520, 370248450, 969323028, 6643838878, 119218851370, 2139295485798, 5600748293800, 100501350283428, 688846502588398, 1803423556807920, 32361122672259148

Offset: 1

Klaus Brockhaus, May 29 2009

nonn

Cf. A096539, A034807, A180363.

[ &+[ Binomial(n-k, k)+Binomial(n-k-1, k-1): k in [1..n div 2] ]: n in [1..80] | IsPrime(n) ];

a(n) = A180363(n) - 1. - Jon Maiga, Jul 24 2019

A180367 a(n) = Lucas(prime(n+1)) - prime(Lucas(n)), for Lucas numbers beginning at 2.

Original entry on oeis.org

0, 2, 6, 22, 182, 490, 3510, 9240, 63868, 1149468, 3009672, 54017304, 370246314, 969319296, 6643832358, 119218840092, 2139295466336, 5600748260454, 100501350226466, 688846502491240, 1803423556642478, 32361122671978600, 221806434537503870, 3980154972736116440

Offset: 0

Jonathan Vos Post, Aug 31 2010

nonn

Commutator of Primes and Lucas numbers. Some subtlety in indexing -- should we start with 0th Lucas number is 2, and 0th prime is 1? As shown here, I use "first" to mean the initial value as shown in P(n) and L(n), even though their indexing differs. This is to A093062 Fibonacci(prime(i))-prime(Fibonacci(i)) as Fibonacci is to Lucas.

			a(0) = 0 because the 1st prime is 2, and the third Lucas number is A000032(2) = 3; while the 1st Lucas number is 2, and the 2nd prime is 3; with 3-3=0.
a(1) = 2 because the 2nd prime is 3, and A000032(3) = 4; while the 2nd Lucas number is 1, and the first2 prime is 2; with 4-2=2.
a(2) = 6 because the 3rd prime is 5, and the 6th Lucas number (counting "2" as first) is A000032(5) = 11; while the 3rd Lucas number is 3, and the 3rd prime is 5; with 11-5=6.
a(3) = 29 - 7 = 22. a(4) = 199 - 17 = 182.

Amiram Eldar, Table of n, a(n) for n = 0..82 (terms 0..50 from Harvey P. Dale)

Cf. A000032, A000040, A094894, A180363, A093062.

A000032 := proc(n) option remember; if n <= 1 then op(n+1,[2,1]) ; else procname(n-1)+procname(n-2) ; end if; end proc:
A094894 := proc(n) ithprime(A000032(n)) ; end proc:
A180363 := proc(n) A000032(ithprime(n)) ; end proc:
A180367 := proc(n) A180363(n+1)-A094894(n) ; end proc: seq(A180367(n),n=0..25) ; # R. J. Mathar, Sep 01 2010

Table[LucasL[Prime[n+1]]-Prime[LucasL[n]],{n,0,30}] (* Harvey P. Dale, Jan 01 2021 *)

a(n) = L(prime(n+1)) - prime(L(n)) = A000032(A000040(n+1)) - A000040(A000032(n)) = A180363(n+1) - A094894(n).

Some indices corrected, 3 values corrected, and formulas signs swapped by R. J. Mathar, Sep 01 2010

A279482 Sum of the first n Lucas numbers whose indices are prime.

Original entry on oeis.org

3, 7, 18, 47, 246, 767, 4338, 13687, 77766, 1227617, 4237966, 58256487, 428504938, 1397827967, 8041666846, 127260518217, 2266556004016, 7867304297817, 108368654581246, 797215157169645, 2600638713977566, 34961761386236715, 256768195924215394

Offset: 1

Vincenzo Librandi, Dec 13 2016

nonn

Partial sums of A180363.

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Cf. A000032, A000040, A111136, A180363.

[&+[Lucas(NthPrime(i)): i in [1..n]]: n in [1..30]];

Mathematica
```
Accumulate[LucasL[Prime[Range[25]]]]
```

From Jon E. Schoenfield, Dec 18 2016: (Start)
a(n) = Sum_{i=1..n} A180363(i).
a(n) = Sum_{i=1..n} A000032(A000040(i)). (End)

cp's OEIS Frontend

A285992 Primes in the bisected Lucas sequence A002878.

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A215809 Prime numbers n for which the Lucas number L(n) (see A000032) is the sum of two squares.

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A215907 Odd numbers n such that the Lucas number L(n) is the sum of two squares.

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A268478 L(p) modulo p^2, where p = prime(n) and L is a Lucas number (A000032).

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A160909 Row sums of triangle defined in A096539.

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A180367 a(n) = Lucas(prime(n+1)) - prime(Lucas(n)), for Lucas numbers beginning at 2.

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A279482 Sum of the first n Lucas numbers whose indices are prime.

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