cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A182504 Composite numbers k that divide both Fibonacci(k+1) and Fibonacci(2k+1)-1.

Original entry on oeis.org

323, 377, 3827, 5777, 6479, 10877, 11663, 18407, 19043, 20999, 23407, 25877, 27323, 34943, 35207, 39203, 44099, 47519, 50183, 51983, 53663, 60377, 65471, 75077, 78089, 79547, 80189, 81719, 82983, 84279, 84419, 86063, 90287, 94667, 100127, 104663, 109871
Offset: 1

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Author

Gary Detlefs, May 05 2012

Keywords

Comments

A subset of A182554 based on a refinement of the Fibonacci criterion for primality described there. The additional constraint that k divides Fibonacci(2*k+1)-1 is suggested by the Cloitre comment in A003631.
What base-2 pseudoprimes are contained in this sequence?
An almost identical sequence can be obtained by testing for composite numbers for which (1) k divides Fibonacci(k+1) and (2) k^12 mod 210 = 1. All primes greater than 7 appear to satisfy condition 2. Terms of {a(n)} which are not pseudoprimes to this criterion are 50183, 65471, 82983, and 84279. - Gary Detlefs, Jun 04 2012

Links

Crossrefs

Cf. A182554, A003631, A038872, A000040.

Programs

  • Magma
    [n: n in [4..11*10^4] | not IsPrime(n) and IsDivisibleBy(Fibonacci(n+1),n) and IsDivisibleBy(Fibonacci(2*n+1)-1,n)]; // Bruno Berselli, May 04 2012
  • Maple
    with (combinat): f:= n-> fibonacci(n): for n from 2 to 100000 do if not isprime(n) and irem(f(n+1), n)=0 and irem((f(2*n+1)-1), n)=0 then print(n) fi od;
  • Mathematica
    Select[Range[110000],CompositeQ[#]&&Mod[Fibonacci[#+1],#]==Mod[Fibonacci[ 2#+1]-1,#] == 0&] (* Harvey P. Dale, Aug 02 2024 *)

A215101 Primes congruent to {2, 3} mod 19.

Original entry on oeis.org

2, 3, 41, 59, 79, 97, 173, 193, 211, 269, 307, 383, 401, 421, 439, 743, 839, 857, 877, 953, 971, 991, 1009, 1123, 1181, 1237, 1409, 1427, 1447, 1523, 1579, 1637, 1693, 1789, 1979, 1997, 2017, 2111, 2131, 2207, 2339, 2377, 2473, 2549, 2663, 2719, 2777, 2833, 2909
Offset: 1

Views

Author

Vincenzo Librandi, Aug 04 2012

Keywords

Links

Crossrefs

Cf. A000040, A003631, A045343, A214888, A214889, A214890.

Programs

  • Magma
    [p: p in PrimesUpTo(3000) | p mod 19 in [2, 3]];
  • Mathematica
    Select[Prime[Range[3000]],MemberQ[{2,3},Mod[#,19]]&]
Select[Flatten[#+{2,3}&/@(19*Range[0,200])],PrimeQ] (* Harvey P. Dale, Oct 07 2019 *)

A344685 Triangle T(n, k) obtained from the array N1(a, b) = a^2 + a*b - b^2, for a >= 0 and b >= 0, read by upwards antidiagonals.

Original entry on oeis.org

0, 1, -1, 4, 1, -4, 9, 5, -1, -9, 16, 11, 4, -5, -16, 25, 19, 11, 1, -11, -25, 36, 29, 20, 9, -4, -19, -36, 49, 41, 31, 19, 5, -11, -29, -49, 64, 55, 44, 31, 16, -1, -20, -41, -64, 81, 71, 59, 45, 29, 11, -9, -31, -55, -81, 100, 89, 76, 61, 44, 25, 4, -19, -44, -71, -100
Offset: 0

Views

Author

Wolfdieter Lang, Jun 17 2021

Keywords

Comments

The general array N(a, b) gives the norms of the integers alpha = a*1 + b*phi, for rational integers a and b, with phi = (1 + sqrt(5))/2 = A001622, in the real quadratic number field Q(phi), also called Q(sqrt(5)). N(a, b) := alpha*alpha' = a^2 + a*b - b^2, with alpha' = a*1 + b*phi' = (a+b)*1 - b*phi. (phi' = (1 - sqrt(5))/2 = 1 - phi = -1/phi.)
The present array is N1(a, b) = N(a, b) = N(-a, -b), for a >= 0 and b >= 0. The companion array N2(a, b) = N(a, -b) = N(-a, b), for a >= 0 and b >= 0 is given (as triangle) in A281386.
The subtriangle N(a, b), with 0 <= b <= a, is given in A281385.
The units u = a + b*phi of the integer domain of Q(phi) satisfy N(a, b) = +1 or -1, and they are related to positive and negative integer powers of phi, involving neighboring Fibonacci numbers a and b of different signs. See, e.g., Hardy and Wright, Theorem 257, p. 222 (units are there called unities).
If |N(alpha)| = q, with q a rational prime, then alpha is a prime in Q(phi). See, e.g., the Dodd reference, Theorem 3.4, p. 23. But there are other primes. For all primes see e.g., Hardy and Wright, Theorem 257, p. 222, or Dodd, Theorem 3.10, p. 25. For rational primes which are also primes in Q(phi) (so-called inert primes) see A003631. See the tables in Appendix B of Dodd, pp. 128 - 150, for the cases p, (p, 0), for all rational primes <= 32717.

Examples

			The array N1(a, b) begins:
a \ b  0   1   2   3   4   5   6   7   8   9   10 ...
-----------------------------------------------------
0:     0  -1  -4  -9 -16 -25 -36 -49 -64 -81 -100 ...
1:     1   1  -1  -5 -11 -19 -29 -41 -55 -71  -89 ...
2:     4   5   4   1  -4 -11 -20 -31 -44 -59  -76 ...
3:     9  11  11   9   5  -1  -9 -19 -31 -45  -61 ...
4:    16  19  20  19  16  11   4  -5 -16 -29  -44 ...
5:    25  29  31  31  29  25  19  11   1 -11  -25 ...
6:    36  41  44  45  44  41  36  29  20   9   -4 ...
7:    49  55  59  61  61  59  55  49  41  31   19 ...
8:    64  71  76  79  80  79  76  71  64  55   44 ...
9:    81  89  95  99 101 101  99  95  89  81   71 ...
10:  100 109 116 121 124 125 124 121 116 109  100 ...
...
-----------------------------------------------------
The Triangle T(n, k) begins:
n \ k  0  1  2  3   4   5   6   7   8   9   10 ...
0:     0
1:     1 -1
2:     4  1 -4
3:     9  5 -1 -9
4:    16 11  4 -5 -16
5:    25 19 11  1 -11 -25
6:    36 29 20  9  -4 -19 -36
7:    49 41 31 19   5 -11 -29 -49
8:    64 55 44 31  16  -1 -20 -41 -64
9:    81 71 59 45  29  11  -9 -31 -55 -81
10:  100 89 76 61  44  25   4 -19 -44 -71 -100
...
------------------------------------------------
Units from norm N(a, b) = N1(a, b) = +1 or -1, for a >= 0 and b >= 0: +(a, b) or -(a, b), with (a, b) = (0, 1), (1, 0), (1, 1), (1, 2), (2, 3), (3, 5), (5, 8), ...; cases + or - phi^n, n >= 0.
Some primes im Q(phi) from |N1(a, b)| = q, with q a prime in Q:
a = 1:  (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 9), (1, 10), ...
a = 2:  (2, 1), (2, 5), (2, 7), (2, 9), ...
a = 3:  (3, 1), (3, 2), (3, 4), (3, 7), (3, 8), (3, 10), ...
a = 4:  (4, 1), (4, 3), (4, 5), (4, 7), (4, 9), ...
a = 5:  (5, 1), (5, 2), (5, 3), (5, 4), (5, 6), (5, 7), (5, 9), ...
a = 6:  (6, 1), (6, 5), (6, 7), ...
a = 7:  (7, 2), (7, 3), (7, 4), (7, 5), (7, 8), (7, 9), (7, 10), ...
a = 8:  (8, 1), (8, 3), (8, 5), (8, 7), ...
a = 9:  (9, 1), (9, 4), (9, 5), (9, 8), (9, 10), ...
a = 10: (10, 1), (10, 9) ...
...

References

  • F. W. Dodd, Number theory in the quadratic field with golden section unit, Polygonal Publishing House, Passaic, NJ.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Clarendon Press Oxford, 2003.

Crossrefs

Cf. A003631, A281385, A132111, A281385, A344686.

Formula

Array N1(a, b) = a^2 + a*b - b^2, for a >= 0 and b >= 0.
Triangle T(n, k) = N1(n-k, k) = n^2 - n*k - k^2, for n >= 0 and k = 0, 1, ..., n.
G.f. for row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k, i.e., g.f. of the triangle: G(x, y) = x*(1 - y + (1 + y - y^2)*x - 2*y*(2 - y)*x^2 + y^2*x^3)/((1 - x*y)^3*(1 - x)^3) (compare with the g.f. in A281385).
Previous Showing 41-43 of 43 results.

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