A082411 -id:A082411 - cp's OEIS frontend

A083216 Fibonacci-like sequence of composite numbers with a(0) = 20615674205555510, a(1) = 3794765361567513.

20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536, 52615644495813559, 80820849424504095, 133436493920317654, 214257343344821749, 347693837265139403, 561951180609961152, 909645017875100555, 1471596198485061707, 2381241216360162262

Offset: 0

Harry J. Smith, Apr 23 2003

a(0) = 20615674205555510, a(1) = 3794765361567513. This is a second-order linear recurrence sequence with a(0) and a(1) coprime that does not contain any primes. It was found by Herbert S. Wilf in 1990.

Seiichi Manyama, Table of n, a(n) for n = 0..4709 (terms 0..1000 from Alois P. Heinz)
Arturas Dubickas, Aivaras Novikas and Jonas Šiurys, A binary linear recurrence sequence of composite numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1737-1749.
R. L. Graham, A Fibonacci-Like sequence of composite numbers, Math. Mag. 37 (1964) 322-324
D. Ismailescu and J. Son, A New Kind of Fibonacci-Like Sequence of Composite Numbers, J. Int. Seq. 17 (2014) # 14.8.2.
Tanya Khovanova, Recursive Sequences
D. E. Knuth, A Fibonacci-Like sequence of composite numbers, Math. Mag. 63 (1) (1990) 21-25
J. W. Nicol, A Fibonacci-like sequence of composite numbers, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R44.
Herbert S. Wilf, Letters to the Editor, Math. Mag. 63, 284, 1990.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A083103, A083104, A083105, A082411, A221286.

a:= n-> (<<0|1>, <1|1>>^n. <<20615674205555510, 3794765361567513>>)[1, 1]:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 04 2013

LinearRecurrence[{1,1},{20615674205555510,3794765361567513},25] (* Paolo Xausa, Nov 07 2023 *)

Vec((20615674205555510-16820908843987997*x)/(1-x-x^2)+O(x^9)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = a(n-1) + a(n-2) for n>1.

G.f.: (20615674205555510-16820908843987997*x)/(1-x-x^2).

A083104 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2), with initial terms 331635635998274737472200656430763, 1510028911088401971189590305498785.

Original entry on oeis.org

331635635998274737472200656430763, 1510028911088401971189590305498785, 1841664547086676708661790961929548, 3351693458175078679851381267428333, 5193358005261755388513172229357881, 8545051463436834068364553496786214

Offset: 0

Harry J. Smith, Apr 23 2003

This is a second-order linear recurrence sequence with a(0) and a(1) coprime that does not contain any primes. It was found by Ronald Graham in 1964.

Indranil Ghosh, Table of n, a(n) for n = 0..4618
Arturas Dubickas, Aivaras Novikas and Jonas Šiurys, A binary linear recurrence sequence of composite numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1737-1749.
R. L. Graham, A Fibonacci-Like sequence of composite numbers, Math. Mag. 37 (1964) 322-324.
D. Ismailescu and J. Son, A New Kind of Fibonacci-Like Sequence of Composite Numbers, J. Int. Seq. 17 (2014) # 14.8.2.
Tanya Khovanova, Recursive Sequences
D. E. Knuth, A Fibonacci-Like sequence of composite numbers, Math. Mag. 63 (1) (1990) 21-25
J. W. Nicol, A Fibonacci-like sequence of composite numbers, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R44.
Carlos Rivera, Problem 31. Fibonacci- all composites sequence, The Prime Puzzles and Problems Connection.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032 (Lucas numbers), A000045 (Fibonacci numbers), A083103, A083105, A083216, A082411, A221286.

LinearRecurrence[{1,1},{331635635998274737472200656430763,1510028911088401971189590305498785},7] (* Harvey P. Dale, Oct 29 2016 *)

a(n)=331635635998274737472200656430763*fibonacci(n-1)+ 1510028911088401971189590305498785*fibonacci(n) \\ Charles R Greathouse IV, Dec 18 2014

G.f.: (331635635998274737472200656430763+1178393275090127233717389649068022*x)/(1-x-x^2). - Colin Barker, Jun 19 2012

Name clarified by Robert C. Lyons, Feb 07 2025

A083105 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2), with initial terms 62638280004239857, 49463435743205655.

Original entry on oeis.org

62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167, 273666867238096679, 435232018728747846, 708898885966844525, 1144130904695592371, 1853029790662436896, 2997160695358029267, 4850190486020466163, 7847351181378495430

Offset: 0

Harry J. Smith, Apr 23 2003

a(0) = 62638280004239857, a(1) = 49463435743205655. This is a second-order linear recurrence sequence with a(0) and a(1) coprime that does not contain any primes. It was found by D. E. Knuth in 1990.

Seiichi Manyama, Table of n, a(n) for n = 0..4705
Arturas Dubickas, Aivaras Novikas and Jonas Šiurys, A binary linear recurrence sequence of composite numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1737-1749.
R. L. Graham, A Fibonacci-Like sequence of composite numbers, Math. Mag. 37 (1964) 322-324.
D. Ismailescu and J. Son, A New Kind of Fibonacci-Like Sequence of Composite Numbers, J. Int. Seq. 17 (2014) # 14.8.2.
Tanya Khovanova, Recursive Sequences
D. E. Knuth, A Fibonacci-like sequence of composite numbers, Math. Mag. 63 (1) (1990) 21-25.
J. W. Nicol, A Fibonacci-like sequence of composite numbers, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R44.
Carlos Rivera, Problem 31. Fibonacci- all composites sequence, The Prime Puzzles and Problems Connection.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032 (Lucas numbers), A000045 (Fibonacci numbers), A083103, A083104, A083216, A082411, A221286.

a:= n-> (<<0|1>, <1|1>>^n. <<62638280004239857, 49463435743205655>>)[1, 1]:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 20 2021

LinearRecurrence[{1,1},{62638280004239857,49463435743205655},20] (* Paolo Xausa, Nov 07 2023 *)

G.f.: (62638280004239857-13174844261034202*x)/(1-x-x^2). [Colin Barker, Jun 19 2012]

Name clarified by Robert C. Lyons, Feb 07 2025

A221286 Vsemirnov's sequence.

Original entry on oeis.org

106276436867, 35256392432, 141532829299, 176789221731, 318322051030, 495111272761, 813433323791, 1308544596552, 2121977920343, 3430522516895, 5552500437238, 8983022954133, 14535523391371, 23518546345504, 38054069736875, 61572616082379, 99626685819254, 161199301901633, 260825987720887, 422025289622520

Offset: 0

Charles R Greathouse IV, Feb 05 2013

A primefree linear recurrence with no common factors. As of 2004 no such sequences with smaller starting terms were known.

Seiichi Manyama, Table of n, a(n) for n = 0..4733 (terms 0..1000 from Alois P. Heinz)
Arturas Dubickas, Aivaras Novikas, and Jonas Šiurys, A binary linear recurrence sequence of composite numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1737-1749.
D. Ismailescu and J. Son, A New Kind of Fibonacci-Like Sequence of Composite Numbers, J. Int. Seq. 17 (2014) # 14.8.2.
Maxim Vsemirnov, A new Fibonacci-like sequence of composite numbers, Journal of Integer Sequences 7:3 (2004).
Index entries for linear recurrences with constant coefficients, signature (1,1).

Other primefree linear recurrences: A083104 (Graham 1964), A082411 (Nicol 1999), A083105 (Knuth 1990), A083216 (Wilf 1990).

a:= n-> (<<0|1>, <1|1>>^n. <<106276436867, 35256392432>>)[1, 1]:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 04 2013

LinearRecurrence[{1, 1}, {106276436867, 35256392432}, 20] (* Alonso del Arte, Feb 05 2013 *)

Vec((106276436867-71020044435*x)/(1-x-x^2)+O(x^30)) \\ Charles R Greathouse IV, Dec 09 2014

a(n) = a(n-1) + a(n-2).

G.f.: (106276436867-71020044435*x)/(1-x-x^2).

Offset corrected by Alois P. Heinz, Apr 04 2013

A347904 Array read by antidiagonals, m, n >= 1: T(m,n) is the first prime (after the two initial terms) in the Fibonacci-like sequence with initial terms m and n, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 3, 7, 0, 5, 5, 5, 5, 5, 11, 0, 0, 0, 7, 7, 7, 7, 7, 7, 7, 23, 0, 13, 0, 11, 0, 17, 17, 41, 0, 23, 13, 0, 11, 19, 19, 0, 17, 0, 0, 0, 13, 0, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 23, 0, 0, 0, 19, 0, 17, 0, 0, 0, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 1

Pontus von Brömssen, Sep 18 2021

There are cases where T(m,n) = 0 even when m and n are coprime; see A082411, A083104, A083105, A083216, and A221286. The smallest (in the sense that m+n is as small as possible) known case where this occurs appears to be m = 106276436867, n = 35256392432 (Vsemirnov's sequence, A221286).

			Array begins:
  m\n|  1  2  3  4  5  6  7  8  9 10  11 12  13  14 15 16
  ---+---------------------------------------------------
   1 |  2  3  7  5 11  7 23 17 19 11  23 13  41  29 31 17
   2 |  3  0  5  0  7  0 41  0 11  0  13  0  43   0 17  0
   3 |  5  5  0  7 13  0 17 11  0 13 103  0  29  17  0 19
   4 |  5  0  7  0 23  0 11  0 13  0  41  0  17   0 19  0
   5 |  7  7 11 13  0 11 19 13 23  0  43 17  31  19  0 37
   6 |  7  0  0  0 11  0 13  0  0  0  17  0  19   0  0  0
   7 | 17 11 13 11 17 13  0 23 41 17  29 19  53   0 37 23
   8 | 19  0 11  0 13  0 37  0 17  0  19  0  89   0 23  0
   9 | 11 11  0 13 19  0 23 17  0 19  31  0 149  23  0 41
  10 | 11  0 13  0  0  0 17  0 19  0  53  0  23   0  0  0
  11 | 13 13 17 19 37 17 43 19 29 31   0 23  37 103 41 43
  12 | 13  0  0  0 17  0 19  0  0  0  23  0 101   0  0  0
  13 | 29 17 19 17 23 19 47 29 31 23  59 37   0  41 43 29
  14 | 31  0 17  0 19  0  0  0 23  0  61  0  67   0 29  0
  15 | 17 17  0 19  0  0 29 23  0  0  37  0  41  29  0 31
  16 | 17  0 19  0 47  0 23  0 59  0 103  0  29   0 31  0
T(2,7) = 41, because the first prime in A022113, excluding the two initial terms, is 41.

Cf. A022113, A082411, A083104, A083105, A083216, A221286, A347905.

# Note that in the (rare) case when m and n are coprime but there are no primes in the Fibonacci-like sequence, this function will go into an infinite loop.
from sympy import isprime,gcd
def A347904(m,n):
    if gcd(m,n) != 1:
        return 0
    m,n = n,m+n
    while not isprime(n):
        m,n = n,m+n
    return n

T(m,n) = 0 if m and n have a common factor.

T(m,n) = T(n,m+n) if m+n is not prime, otherwise T(m,n) = m+n.

A347905 Array read by antidiagonals, m, n >= 1: T(m,n) is the position of the first prime (after the two initial terms) in the Fibonacci-like sequence with initial terms m and n, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 2, 3, 0, 3, 2, 2, 2, 2, 3, 0, 0, 0, 3, 2, 2, 2, 2, 2, 2, 4, 0, 3, 0, 3, 0, 4, 3, 5, 0, 4, 3, 0, 3, 4, 3, 0, 3, 0, 0, 0, 3, 0, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 0, 6, 0, 3, 0, 0, 0, 3, 0, 3, 0, 4

Offset: 1

Pontus von Brömssen, Sep 18 2021

There are cases where T(m,n) = 0 even when m and n are coprime; see A082411, A083104, A083105, A083216, and A221286.

The largest value of T(m,n) for m, n <= 5000 is T(1591,300) = 17262.

			Array begins:
  m\n|  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
  ---+------------------------------------------------------------
   1 |  2  2  3  2  3  2  4  3  3  2  3  2  4  3  3  2  4  2  4  3
   2 |  2  0  2  0  2  0  5  0  2  0  2  0  4  0  2  0  2  0  4  0
   3 |  3  2  0  2  3  0  3  2  0  2  6  0  3  2  0  2  3  0  3  2
   4 |  2  0  2  0  4  0  2  0  2  0  4  0  2  0  2  0  4  0  2  0
   5 |  3  2  3  3  0  2  3  2  3  0  4  2  3  2  0  3  4  2  3  0
   6 |  2  0  0  0  2  0  2  0  0  0  2  0  2  0  0  0  2  0  5  0
   7 |  4  3  3  2  3  2  0  3  4  2  3  2  4  0  3  2  3  3  4  3
   8 |  4  0  2  0  2  0  4  0  2  0  2  0  5  0  2  0  4  0  4  0
   9 |  3  2  0  2  3  0  3  2  0  2  3  0  6  2  0  3  3  0  3  2
  10 |  2  0  2  0  0  0  2  0  2  0  4  0  2  0  0  0  4  0  2  0
  11 |  3  2  3  3  4  2  4  2  3  3  0  2  3  5  3  3  4  2  4  2
  12 |  2  0  0  0  2  0  2  0  0  0  2  0  5  0  0  0  2  0  2  0
  13 |  4  3  3  2  3  2  4  3  3  2  4  3  0  3  3  2  3  2  4  3
  14 |  4  0  2  0  2  0  0  0  2  0  4  0  4  0  2  0  2  0  5  0
  15 |  3  2  0  2  0  0  3  2  0  0  3  0  3  2  0  2  6  0  3  0
  16 |  2  0  2  0  4  0  2  0  4  0  5  0  2  0  2  0  4  0  4  0
  17 |  3  2  3  5 10  2  3  6  4  3  4  2  3  2  3  5  0  3  7  2
  18 |  2  0  0  0  2  0  5  0  0  0  2  0  2  0  0  0  5  0  2  0
  19 |  4  3  4  2  3  3  4  5  3  2  3  2  6  3  4  5  3  2  0  3
  20 |  4  0  2  0  0  0  4  0  2  0  2  0  4  0  0  0  2  0  4  0
T(2,7) = 5, because 5 is the smallest k >= 2 for which A022113(k) is prime.

Cf. A022113, A082411, A083104, A083105, A083216, A221286, A347904.

# Note that in the (rare) case when m and n are coprime but there are no primes in the Fibonacci-like sequence, this function will go into an infinite loop.
from sympy import isprime,gcd
def A347905(m,n):
    if gcd(m,n) != 1:
        return 0
    m,n = n,m+n
    k=2
    while not isprime(n):
        m,n = n,m+n
        k += 1
    return k

T(m,n) = 0 if m and n have a common factor.

T(m,n) = T(n,m+n) + 1 if m+n is not prime, otherwise T(m,n) = 2.

A348204 Fibonacci-like sequence of composite numbers with a(0) = 759135467284, a(1) = 74074527465.

Original entry on oeis.org

759135467284, 74074527465, 833209994749, 907284522214, 1740494516963, 2647779039177, 4388273556140, 7036052595317, 11424326151457, 18460378746774, 29884704898231, 48345083645005, 78229788543236, 126574872188241, 204804660731477, 331379532919718, 536184193651195

Offset: 0

Chittaranjan Pardeshi, Oct 06 2021

This is a second-order linear recurrence sequence with a(0) and a(1) coprime that does not contain any primes.

This sequence was found using Knuth's method.

R. L. Graham, A Fibonacci-Like sequence of composite numbers, Math. Mag., Vol. 37, No. 5 (1964), pp. 322-324.
Donald E. Knuth, A Fibonacci-Like sequence of composite numbers, Math. Mag., Vol. 63, No. 1 (1990), pp. 21-25.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A221286, A083104, A082411, A083105, A083216, A083216.

a:= n-> (<<0|1>, <1|1>>^n. <<759135467284, 74074527465>>)[1, 1]:
seq(a(n), n=0..16);  # Alois P. Heinz, Oct 06 2021

LinearRecurrence[{1, 1}, {759135467284, 74074527465}, 17] (* Amiram Eldar, Oct 07 2021 *)

a(n)=759135467284*fibonacci(n-1)+ 74074527465*fibonacci(n)

a(n) = a(n-1) + a(n-2).

cp's OEIS Frontend

A083216 Fibonacci-like sequence of composite numbers with a(0) = 20615674205555510, a(1) = 3794765361567513.

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A083104 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2), with initial terms 331635635998274737472200656430763, 1510028911088401971189590305498785.

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A083105 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2), with initial terms 62638280004239857, 49463435743205655.

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A221286 Vsemirnov's sequence.

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A347904 Array read by antidiagonals, m, n >= 1: T(m,n) is the first prime (after the two initial terms) in the Fibonacci-like sequence with initial terms m and n, or 0 if no such prime exists.

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A347905 Array read by antidiagonals, m, n >= 1: T(m,n) is the position of the first prime (after the two initial terms) in the Fibonacci-like sequence with initial terms m and n, or 0 if no such prime exists.

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A348204 Fibonacci-like sequence of composite numbers with a(0) = 759135467284, a(1) = 74074527465.

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