A175723 -id:A175723 - cp's OEIS frontend

A214897 Conway's subprime Fibonacci sequence: cycle lengths.

1, 10, 11, 18, 19, 56, 136

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with a pair of positive integers, but each new nonprime term is divided by its least prime factor. The recursion enters a loop of length a(n) after a finite number of steps. Conjecture: the list of loops is complete (checked to [10^5, 10^5]), loops of length a(n) are unique and no infinite chains exist.

Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099v1 [math.NT]

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

Mathematica
```
see A214674
```

A214893 Conway's subprime Fibonacci sequence starting with (18, 5).

Original entry on oeis.org

18, 5, 23, 14, 37, 17, 27, 22, 7, 29, 18, 47, 13, 30, 43, 73, 58, 131, 63, 97, 80, 59, 139, 99, 119, 109, 114, 223, 337, 280, 617, 299, 458, 757, 405, 581, 493, 537, 515, 526, 347, 291, 319, 305, 312, 617, 929

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (18, 5), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 56 after 26 terms on reaching (119, 109).

Wouter Meeussen, Table of n, a(n) for n = 1..82
Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099v1 [math.NT]

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

see A214674
nxt[{a_,b_}]:=Module[{c=a+b},{b,If[PrimeQ[c],c,c/FactorInteger[c][[1,1]]]}]; Transpose[NestList[nxt,{18,5},82]][[1]] (* Harvey P. Dale, Oct 19 2012 *)

A221183 a(0)=0, a(1)=1; thereafter a(n) = gpf(2*a(n-1)+a(n-2)), where gpf = "greatest prime factor" (A006530).

Original entry on oeis.org

0, 1, 2, 5, 3, 11, 5, 7, 19, 5, 29, 7, 43, 31, 7, 5, 17, 13, 43, 11, 13, 37, 29, 19, 67, 17, 101, 73, 19, 37, 31, 11, 53, 13, 79, 19, 13, 5, 23, 17, 19, 11, 41, 31, 103, 79, 29, 137, 101, 113, 109, 331, 257, 13, 283, 193, 223, 71, 73, 31, 5, 41, 29, 11, 17, 5, 3, 11, 5, 7, 19, 5, 29, 7, 43, 31, 7, 5, 17

Offset: 0

Gary W. Adamson and N. J. A. Sloane, Jan 19 2013

nonn

Rapidly enters a loop of length 62: [5, 3, 11, 5, 7, 19, 5, 29, 7, 43, 31, 7, 5, 17, 13, 43, 11, 13, 37, 29, 19, 67, 17, 101, 73, 19, 37, 31, 11, 53, 13, 79, 19, 13, 5, 23, 17, 19, 11, 41, 31, 103, 79, 29, 137, 101, 113, 109, 331, 257, 13, 283, 193, 223, 71, 73, 31, 5, 41, 29, 11, 17].

Greg Back and Mihai Caragiu, The Greatest Prime Factor and Recurrent Sequences, Fibonacci Quart. 48 (2010), no. 4, 358-362. [Discusses similar sequences]

Cf. A006530, A175723.

gpf(n) = if (n==1, 1, vecmax(factor(n)[,1]));
lista(nn) = {print1(x=0, ", "); print1(y=1, ", "); for (n=2, nn, z = gpf(x+2*y); print1(z, ", "); x = y; y = z;);} \\ Michel Marcus, Mar 01 2016

A277110 a(n) = A066048(a(n-1) + a(n-2)) with a(0) = 0 and a(1) = 1.

Original entry on oeis.org

0, 1, 1, 4, 25, 841, 866, 1707, 2573, 214, 2787, 9006001, 6898, 1099, 7997, 758, 515, 1273, 298, 2468041, 2468339, 5674, 2474013, 13853, 2487866, 5357, 2493223, 1322, 498909, 250231053361, 7149472922, 203937, 420569227, 1581854, 9201, 1591055, 94, 13371, 13465, 13418, 309, 371, 34, 15, 49, 4, 2809

Offset: 0

Altug Alkan, Feb 09 2017

nonn

The sequence enters a cycle after 1440 terms.

7833343487493084437^2 is the largest term of this sequence.

			a(4) = 25 because A066048(1 + 4) = A066048(5) = 25.

Altug Alkan, Table of n, a(n) for n = 0..2000

Cf. A000045, A066048, A175723.

a(n) = a(n+480) for n > 1440.

A287051 a(0) = 0, a(1) = 1; a(2n) = gpf(a(n)), a(2n+1) = a(n) + a(n+1), where gpf(a(n)) is the greatest prime dividing a(n) for a(n) >= 2 and 1 if a(n) = 1 (A006530).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 2, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 2, 5, 1, 6, 5, 7, 2, 9, 7, 10, 3, 11, 2, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 2, 11, 3, 10, 7, 9, 2, 7, 5, 6, 1, 7, 3, 11, 5, 12, 7, 9, 2, 11, 3, 16, 7, 17, 5, 13, 3, 14, 11, 13, 2, 15, 13, 18, 5, 17, 3, 19, 7, 16, 3, 11, 2, 11, 3, 16, 7

Offset: 0

Ilya Gutkovskiy, May 18 2017

nonn

A variation on Stern's diatomic sequence.

			a(0) = 0;
a(1) = 1;
a(2) = a(2*1) = gpf(a(1)) = 1;
a(3) = a(2*1+1) = a(1) + a(2) = 2;
a(4) = a(2*2) = gpf(a(2)) = 1;
a(5) = a(2*2+1) = a(2) + a(3) = 3;
a(6) = a(2*3) = gpf(a(3)) = 2, etc.

Michael Gilleland, Some Self-Similar Integer Sequences
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Eric Weisstein's World of Mathematics, Stern's Diatomic Series
Index entries for sequences related to Stern's sequences

Cf. A002487, A006530, A175723, A177904.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], FactorInteger[a[n/2]][[-1, 1]], a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 100}]

A361520 a(n) is the greatest prime factor of a(n-2)^2 + a(n-1)^2 where a(1)=2 and a(2)=3.

Original entry on oeis.org

2, 3, 13, 89, 809, 349, 409, 144541, 10446133981, 1361264878245241, 4398505263882824939701, 17847523009215848981, 512996953133650208042047593649109478833

Offset: 1

Sebastian F. Orellana, Mar 14 2023

nonn

Sebastian F. Orellana, Table of n, a(n) for n = 1..15

Cf. A006530, A113494, A175723.

A[1]:= 2: A[2]:= 3:
for n from 3 to 15 do A[n]:= max(numtheory:-factorset(A[n-2]^2 + A[n-1]^2)) od:
seq(A[],n=1..15); # Robert Israel, Mar 17 2023

a[1] = 2; a[2] = 3; a[n_] := a[n] = FactorInteger[a[n - 1]^2 + a[n - 2]^2][[-1, 1]]; Array[a, 14] (* Amiram Eldar, Mar 14 2023 *)

A214894 Conway's subprime Fibonacci sequence starting with (10, 18).

Original entry on oeis.org

10, 18, 14, 16, 15, 31, 23, 27, 25, 26, 17, 43, 30, 73, 103, 88, 191, 93, 142, 47, 63, 55, 59, 57, 58, 23, 27

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (10, 18), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 19 after 8 terms on reaching (23, 27).

Richard K. Guy, Tanya Khovanova, and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099 [math.NT]

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

Mathematica
```
see A214674
```

A214895 Conway's subprime Fibonacci sequence starting with (23, 162).

Original entry on oeis.org

23, 162, 37, 199, 118, 317, 145, 231, 188, 419, 607, 513, 560, 37, 199

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (23, 162), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 11 after 4 terms on reaching (37, 199).

Richard K. Guy, Tanya Khovanova, and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099 [math.NT]

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

Mathematica
```
see A214674
```

A214896 Conway's subprime Fibonacci sequence starting with (382, 127).

Original entry on oeis.org

382, 127, 509, 318, 827, 229, 528, 757, 257, 507, 382, 127

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (382, 127), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 10 after 2 terms on reaching (382, 127).

Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099v1 [math.NT]

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

Mathematica
```
see A214674
```

cp's OEIS Frontend

A214897 Conway's subprime Fibonacci sequence: cycle lengths.

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A214893 Conway's subprime Fibonacci sequence starting with (18, 5).

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A221183 a(0)=0, a(1)=1; thereafter a(n) = gpf(2*a(n-1)+a(n-2)), where gpf = "greatest prime factor" (A006530).

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PARI

A277110 a(n) = A066048(a(n-1) + a(n-2)) with a(0) = 0 and a(1) = 1.

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A287051 a(0) = 0, a(1) = 1; a(2*n) = gpf(a(n)), a(2*n+1) = a(n) + a(n+1), where gpf(a(n)) is the greatest prime dividing a(n) for a(n) >= 2 and 1 if a(n) = 1 (A006530).

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Mathematica

A361520 a(n) is the greatest prime factor of a(n-2)^2 + a(n-1)^2 where a(1)=2 and a(2)=3.

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Maple

Mathematica

A214894 Conway's subprime Fibonacci sequence starting with (10, 18).

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Programs

Mathematica

A214895 Conway's subprime Fibonacci sequence starting with (23, 162).

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Crossrefs

Programs

Mathematica

A214896 Conway's subprime Fibonacci sequence starting with (382, 127).

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A287051 a(0) = 0, a(1) = 1; a(2n) = gpf(a(n)), a(2n+1) = a(n) + a(n+1), where gpf(a(n)) is the greatest prime dividing a(n) for a(n) >= 2 and 1 if a(n) = 1 (A006530).