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Search: a221286 -id:a221286
Displaying 1-7 of 7 results found. page 1
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A083216 Fibonacci-like sequence of composite numbers with a(0) = 20615674205555510, a(1) = 3794765361567513. +10
11
20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536, 52615644495813559, 80820849424504095, 133436493920317654, 214257343344821749, 347693837265139403, 561951180609961152, 909645017875100555, 1471596198485061707, 2381241216360162262 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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.
LINKS
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).
FORMULA
a(n) = a(n-1) + a(n-2) for n>1.
G.f.: (20615674205555510-16820908843987997*x)/(1-x-x^2).
MAPLE
a:= n-> (<<0|1>, <1|1>>^n. <<20615674205555510, 3794765361567513>>)[1, 1]:
seq(a(n), n=0..20); # Alois P. Heinz, Apr 04 2013
MATHEMATICA
LinearRecurrence[{1, 1}, {20615674205555510, 3794765361567513}, 25] (* Paolo Xausa, Nov 07 2023 *)
PROG
(PARI) Vec((20615674205555510-16820908843987997*x)/(1-x-x^2)+O(x^9)) \\ Charles R Greathouse IV, Sep 23 2012
CROSSREFS
Cf. A000032, A000045, A083103, A083104, A083105, A082411, A221286.
KEYWORD
nonn,easy
AUTHOR
Harry J. Smith, Apr 23 2003
STATUS
approved
A083104 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). +10
9
331635635998274737472200656430763, 1510028911088401971189590305498785, 1841664547086676708661790961929548, 3351693458175078679851381267428333, 5193358005261755388513172229357881, 8545051463436834068364553496786214 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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.
LINKS
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).
FORMULA
G.f.: (331635635998274737472200656430763+1178393275090127233717389649068022*x)/(1-x-x^2). - Colin Barker, Jun 19 2012
MATHEMATICA
LinearRecurrence[{1, 1}, {331635635998274737472200656430763, 1510028911088401971189590305498785}, 7] (* Harvey P. Dale, Oct 29 2016 *)
PROG
(PARI) a(n)=331635635998274737472200656430763*fibonacci(n-1)+ 1510028911088401971189590305498785*fibonacci(n) \\ Charles R Greathouse IV, Dec 18 2014
CROSSREFS
Cf. A000032 (Lucas numbers), A000045 (Fibonacci numbers), A083103, A083105, A083216, A082411, A221286.
KEYWORD
nonn,easy
AUTHOR
Harry J. Smith, Apr 23 2003
STATUS
approved
A083105 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). +10
9
62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167, 273666867238096679, 435232018728747846, 708898885966844525, 1144130904695592371, 1853029790662436896, 2997160695358029267, 4850190486020466163, 7847351181378495430 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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.
LINKS
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).
FORMULA
G.f.: (62638280004239857-13174844261034202*x)/(1-x-x^2). [Colin Barker, Jun 19 2012]
MAPLE
a:= n-> (<<0|1>, <1|1>>^n. <<62638280004239857, 49463435743205655>>)[1, 1]:
seq(a(n), n=0..20); # Alois P. Heinz, Sep 20 2021
MATHEMATICA
LinearRecurrence[{1, 1}, {62638280004239857, 49463435743205655}, 20] (* Paolo Xausa, Nov 07 2023 *)
CROSSREFS
Cf. A000032 (Lucas numbers), A000045 (Fibonacci numbers), A083103, A083104, A083216, A082411, A221286.
KEYWORD
nonn,easy
AUTHOR
Harry J. Smith, Apr 23 2003
STATUS
approved
A082411 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). +10
8
407389224418, 76343678551, 483732902969, 560076581520, 1043809484489, 1603886066009, 2647695550498, 4251581616507, 6899277167005, 11150858783512, 18050135950517, 29200994734029, 47251130684546, 76452125418575, 123703256103121, 200155381521696, 323858637624817 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
a(0) = 407389224418, a(1) = 76343678551. 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 John Nicol in 1999.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
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).
FORMULA
G.f.: (407389224418-331045545867*x)/(1-x-x^2). [Colin Barker, Jun 19 2012]
MAPLE
a:= n-> (<<0|1>, <1|1>>^n. <<407389224418, 76343678551>>)[1, 1]:
seq(a(n), n=0..20); # Alois P. Heinz, Apr 04 2013
MATHEMATICA
LinearRecurrence[{1, 1}, {407389224418, 76343678551}, 25] (* Paolo Xausa, Nov 07 2023 *)
CROSSREFS
Cf. A000032 (Lucas numbers), A000045 (Fibonacci numbers), A083103, A083104, A083105, A083216, A221286.
KEYWORD
nonn,easy
AUTHOR
Harry J. Smith, Apr 23 2003
STATUS
approved
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. +10
2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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).
LINKS
Table of n, a(n) for n=1..78.
FORMULA
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.
EXAMPLE
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.
PROG
(Python)
# 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
CROSSREFS
Cf. A022113, A082411, A083104, A083105, A083216, A221286, A347905.
KEYWORD
nonn,tabl
AUTHOR
Pontus von Brömssen, Sep 18 2021
STATUS
approved
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. +10
2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..91.
FORMULA
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.
EXAMPLE
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.
PROG
(Python)
# 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
CROSSREFS
Cf. A022113, A082411, A083104, A083105, A083216, A221286, A347904.
KEYWORD
nonn,tabl
AUTHOR
Pontus von Brömssen, Sep 18 2021
STATUS
approved
A348204 Fibonacci-like sequence of composite numbers with a(0) = 759135467284, a(1) = 74074527465. +10
0
759135467284, 74074527465, 833209994749, 907284522214, 1740494516963, 2647779039177, 4388273556140, 7036052595317, 11424326151457, 18460378746774, 29884704898231, 48345083645005, 78229788543236, 126574872188241, 204804660731477, 331379532919718, 536184193651195 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..16.
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).
FORMULA
a(n) = a(n-1) + a(n-2).
MAPLE
a:= n-> (<<0|1>, <1|1>>^n. <<759135467284, 74074527465>>)[1, 1]:
seq(a(n), n=0..16); # Alois P. Heinz, Oct 06 2021
MATHEMATICA
LinearRecurrence[{1, 1}, {759135467284, 74074527465}, 17] (* Amiram Eldar, Oct 07 2021 *)
PROG
(PARI) a(n)=759135467284*fibonacci(n-1)+ 74074527465*fibonacci(n)
CROSSREFS
Cf. A221286, A083104, A082411, A083105, A083216, A083216.
KEYWORD
nonn,easy
AUTHOR
Chittaranjan Pardeshi, Oct 06 2021
STATUS
approved
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Last modified September 12 19:14 EDT 2024. Contains 375853 sequences. (Running on oeis4.)