A000322 -id:A000322 - cp's OEIS frontend

A253705 Indices of primes in the 8th-order Fibonacci number sequence, A079262.

9, 17, 25, 125, 350, 1322, 108935, 199528

Offset: 1

Robert Price, Jan 09 2015

a(9) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A001631, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523, A214825, A235862, A000288, A000322, A000383, A249413, A060455, A079262.

a={0,0,0,0,0,0,0,1}; step=8; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst
Flatten[Position[LinearRecurrence[{1,1,1,1,1,1,1,1},{0,0,0,0,0,0,0,1},200000],?PrimeQ]]-1 (* The program takes a long time to run *) (* _Harvey P. Dale, Apr 26 2018 *)

lista(nn) = {gf = x^7/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8); for (n=0, nn, if (isprime(polcoeff(gf+O(x^(n+1)), n)), print1(n, ", ")););} \\ Michel Marcus, Jan 12 2015

A163551 13th-order Fibonacci numbers: a(n) = a(n-1) + ... + a(n-13) with a(1)=...=a(13)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196597, 393169, 786289, 1572481, 3144769, 6289153, 12577537, 25153537, 50304001, 100601857, 201191425, 402358273, 804667393

Offset: 1

Jainit Purohit (mjainit(AT)gmail.com), Jul 30 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1,1).

Cf. A000045 (Fibonacci numbers), A000213 (tribonacci), A000288 (tetranacci), A000322 (pentanacci), A000383 (hexanacci), A060455 (heptanacci), A123526 (octanacci), A127193 (nonanacci), A127194 (decanacci), A127624 (undecanacci), A207539 (dodecanacci).

With[{c=Table[1,{13}]},LinearRecurrence[c,c,40]] (* Harvey P. Dale, Aug 09 2013 *)

x='x+O('x^50); Vec((1-x^2 -2*x^3-3*x^4 -4*x^5-5*x^6 -6*x^7-7*x^8 -8*x^9 -9*x^10 -10*x^11 -11*x^12) / (1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10-x^11-x^12-x^13)) \\ G. C. Greubel, Jul 28 2017

a(n) = a(n-1)+a(n-2)+...+a(n-13) for n > 12, a(0)=a(1)=...=a(12)=1.

G.f.: (-1)*(-1+x^2+2*x^3+3*x^4+4*x^5+5*x^6+6*x^7+7*x^8+8*x^9+9*x^10 +10*x^11 +11*x^12) / (1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10-x^11-x^12-x^13). - Michael Burkhart, Feb 18 2012

Values adapted to the definition by R. J. Mathar, Aug 01 2009

A253333 Primes in the 7th-order Fibonacci numbers A060455.

Original entry on oeis.org

7, 13, 97, 193, 769, 1531, 3049, 6073, 12097, 24097, 95617, 379399, 2998753, 187339729, 373174033, 2949551617, 184265983633, 731152932481, 88025699967469825543, 175344042716296888429, 4979552865927484193343796114081304399449

Offset: 1

Robert Price, Dec 30 2014

nonn

a(22) is too large to display here. It has 53 digits and is the 180th term in A060455.

Robert G. Wilson v, Table of n, a(n) for n = 1..34

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920, A000383, A247192, A060455, A253318.

a={1,1,1,1,1,1,1}; step=7; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[7]]=sum]; lst
With[{c=PadRight[{},7,1]},Select[LinearRecurrence[c,c,150],PrimeQ]] (* Harvey P. Dale, May 08 2015 *)

lista(nn) = {gf = ( -1+x^2+2*x^3+3*x^4+4*x^5+5*x^6 ) / ( -1+x+x^2+x^3+x^4+x^5+x^6+x^7 ); for (n=0, nn, if (isprime(p=polcoeff(gf+O(x^(n+1)), n)), print1(p, ", ")););} \\ Michel Marcus, Jan 11 2015

A254412 Indices of primes in the 8th-order Fibonacci number sequence, A123526.

Original entry on oeis.org

11, 13, 15, 24, 30, 33, 57, 104, 121, 132, 149, 158, 178, 220, 295, 389, 1070, 1101, 1373, 1761, 1778, 2333, 2731, 4541, 5189, 5237, 5738, 8025, 8787, 10561, 11783, 13435, 14638, 15337, 20985, 21722, 24770, 31009, 57367, 65877, 129773, 134630, 167020

Offset: 1

Robert Price, Jan 30 2015

a(44) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A001631, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523, A214825, A235862, A000288, A000322, A000383, A249413, A060455, A079262, A123526, A254413.

a={1,1,1,1,1,1,1,1}; step=8; lst={}; For[n=step+1,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst

A254413 Primes in the 8th-order Fibonacci numbers A123526.

Original entry on oeis.org

29, 113, 449, 226241, 14307889, 113783041, 1820091580429249, 233322881089059894782836851617, 29566627412209231076314948970028097, 59243719929958343565697204780596496129, 7507351981539044730893385057192143660843521

Offset: 1

Robert Price, Jan 30 2015

nonn

a(12) is too large to display here. It has 46 digits and is the 158th term in A123526.

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920, A000383, A247192, A060455, A253318, A079262, A253705, A123526, A254412.

a={1,1,1,1,1,1,1,1}; step=8; lst={}; For[n=step+1,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[step]]=sum]; lst
Select[With[{lr=PadRight[{},8,1]},LinearRecurrence[lr,lr,200]],PrimeQ] (* Harvey P. Dale, Dec 03 2022 *)

A105527 Index when n-nacci (tribonacci etc.) passes Fibonacci numbers.

Original entry on oeis.org

3, 4, 6, 9, 11, 14, 16, 19, 22, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 67, 70, 73, 76, 79, 82, 85, 88, 91, 95, 98, 101, 104, 107, 110, 113, 117, 120, 123, 126, 129, 132, 136, 139, 142, 145, 148, 151, 155, 158, 161, 164, 167, 171, 174, 177, 180, 183, 186

Offset: 3

Ferenc Adorjan (fadorjan(AT)freemail.hu), Apr 11 2005

The n-nacci sequences are considered as defined in A000213, A000288, A000322, A000383 for n=3,4,5,6, respectively.

			a(3)=3 because tribonacci(3)=3 while fibonacci(3)=2;
a(6)=9 because hexanacci(9)=41 while fibonacci(9)=34.

Cf. A105528, A000045, A000213, A000288, A000322, A000383.

{n_bonacci(n,v)= local(x,l);l=matsize(v)[2]; x=0;for(i=l-n+1,l,x+=v[i]);return(x)} {nbg(k)= local(q=[],l,v,m); for(n=3,k,v=[];for(i=1,n,v=concat(v,1));m=n; while((x=n_bonacci(n,v))<=fibonacci(m),v=concat(v,x);m+=1);q=concat(q,m)); return(q) }

A105528 First differences of A105527.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3

Offset: 0

Ferenc Adorjan (fadorjan(AT)freemail.hu), Apr 11 2005

Note that if n>9, a(n) is seemingly either 3 or 4. Checked up to index 200.

Cf. A105527, A000045, A000213, A000288, A000322, A000383.

{n_bonacci(n,v)= local(x,l);l=matsize(v)[2]; x=0;for(i=l-n+1,l,x+=v[i]);return(x)} {nbg(k)= local(q=[],l,v,m); for(n=3,k,v=[];for(i=1,n,v=concat(v,1));m=n; while((x=n_bonacci(n,v))<=fibonacci(m),v=concat(v,x);m+=1);q=concat(q,m)); return(q) } z=nbg(100);d=[];for(i=2,98,d=concat(d,z[i]-z[i-1]))

A174323 Numbers n such that omega(Fibonacci(n)) is a square.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 11, 13, 17, 20, 23, 24, 27, 28, 29, 32, 43, 47, 52, 55, 74, 77, 80, 83, 84, 85, 87, 88, 91, 93, 96, 97, 100, 108, 115, 123, 131, 132, 137, 138, 143, 146, 149, 156, 157, 161, 163, 178, 184, 187, 189, 196, 197, 209, 211, 214, 215, 221, 222, 223, 232

Offset: 1

Michel Lagneau, Mar 15 2010

nonn

Numbers n such that omega(A000045(n)) is a square, where omega(p) is the number of distinct prime factors of p (A001221). Remark: for the larger Fibonacci numbers F(n) (n > 300), the Maple program (below) is very slow. So we use a two-step process: factoring F(n) with the elliptic curve method, and then calculate the distinct prime factors.

			omega(Fibonacci(1)) = omega(Fibonacci(2)) = omega(1) = 0,
omega(Fibonacci(3)) = omega(2) = 1,
omega(Fibonacci(20)) = omega(6765) = 4,
omega(Fibonacci(80)) = omega(23416728348467685) = 9.

Majorie Bicknell and Verner E Hoggatt, Fibonacci's Problem Book, Fibonacci Association, San Jose, Calif., 1974.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 1-8.

Amiram Eldar, Table of n, a(n) for n = 1..258 (terms 1..200 from Robert Israel, derived from b-file for A022307)
Blair Kelly, Fibonacci and Lucas Factorizations
Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
Eric Weisstein's World of Mathematics, Fibonacci Number
Wikipedia, Fibonacci number

Cf. A038575 (number of prime factors of n-th Fibonacci number, with multiplicity).
Cf. A000045, A000213, A000288, A000322, A000383, A001221, A060455, A030186, A039834, A020695, A020701, A071679.
Cf. A022307 (number of distinct prime factors of n-th Fibonacci number), A086597 (number of primitive prime factors).

[k:k in [1..240]| IsSquare(#PrimeDivisors(Fibonacci(k)))]; // Marius A. Burtea, Oct 15 2019

with(numtheory):u0:=0:u1:=1:for p from 2 to 400 do :s:=u0+u1:u0:=u1:u1:=s: s1:=nops( ifactors(s)[2]): w1:=sqrt(s1):w2:=floor(w1):if w1=w2 then print (p): else fi:od:
# alternative:
P[1]:= {}: count:= 1: res:= 1:
for i from 2 to 300 do
  pn:= map(t -> i/t, numtheory:-factorset(i));
  Cprimes:= `union`(seq(P[t],t=pn));
  f:= combinat:-fibonacci(i);
  for p in Cprimes do f:= f/p^padic:-ordp(f,p) od;
  P[i]:= Cprimes union numtheory:-factorset(f);
  if issqr(nops(P[i])) then
     count:= count+1;
     res:= res, i;
  fi;
od:
res; # Robert Israel, Oct 13 2016

Select[Range[200], IntegerQ[Sqrt[PrimeNu[Fibonacci[#]]]] &] (* G. C. Greubel, May 16 2017 *)

is(n)=issquare(omega(fibonacci(n))) \\ Charles R Greathouse IV, Oct 13 2016

A248700 Indices of primes in the Heptanacci numbers sequence A122189.

Original entry on oeis.org

8, 14, 22, 102495, 130447, 173590

Offset: 1

Robert Price, Dec 02 2014

a(7) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A001631, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523, A214825, A235862, A000288, A000322, A122189.

a={0,0,0,0,0,0,1}; For[n=7, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[n]]; a=RotateLeft[a]; a[[7]]=sum]

A253706 Primes in the 8th-order Fibonacci numbers A079262.

Original entry on oeis.org

2, 509, 128257, 133294824621464999938178340471931877, 4596852049500861351052672455121859744010232939954169259264638023409631672658340253083284317818242062413

Offset: 1

Robert Price, Jan 09 2015

nonn

a(6) is too large to display here. It has 395 digits and is the 1322nd term in A079262.

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920, A000383, A247192, A060455, A253318, A079262, A253705.

a={0,0,0,0,0,0,0,1}; step=8; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[step]]=sum]; lst

lista(nn) = {gf = x^7/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8); for (n=0, nn, if (isprime(p=polcoeff(gf+O(x^(n+1)), n)), print1(p, ", ")););} \\ Michel Marcus, Jan 12 2015

cp's OEIS Frontend

A253705 Indices of primes in the 8th-order Fibonacci number sequence, A079262.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A163551 13th-order Fibonacci numbers: a(n) = a(n-1) + ... + a(n-13) with a(1)=...=a(13)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A253333 Primes in the 7th-order Fibonacci numbers A060455.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A254412 Indices of primes in the 8th-order Fibonacci number sequence, A123526.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A254413 Primes in the 8th-order Fibonacci numbers A123526.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A105527 Index when n-nacci (tribonacci etc.) passes Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A105528 First differences of A105527.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A174323 Numbers n such that omega(Fibonacci(n)) is a square.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI