A000383 -id:A000383 - cp's OEIS frontend

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

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

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

A255529 Indices of primes in the 9th-order Fibonacci number sequence, A104144.

Original entry on oeis.org

10, 19, 878

Offset: 1

Robert Price, Feb 24 2015

a(4) > 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, A104144.

a={0,0,0,0,0,0,0,0,1}; step=9; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst

a104144(n) = polcoeff(x^8/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9) + O(x^(n+1)), n);
lista(nn) = {for (n=1, nn, if (isprime(a104144(n)), print1(n, ", ")););} \\ Michel Marcus, Feb 27 2015

A255530 Indices of primes in the 9th-order Fibonacci number sequence, A251746.

Original entry on oeis.org

10, 19, 59, 79, 12487

Offset: 1

Robert Price, Feb 24 2015

a(6) > 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, A251746.

a={0,0,0,0,0,0,0,1,0}; step=9; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst

A255531 Indices of primes in the 9th-order Fibonacci number sequence, A251747.

Original entry on oeis.org

10, 16, 116, 236, 316, 1376, 5066, 103696, 120949

Offset: 1

Robert Price, Feb 24 2015

a(10) > 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, A251747.

a={0,0,0,0,0,0,1,0,0}; step=9; 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[Table[1,{9}],{0,0,0,0,0,0,1,0,0},125000],?PrimeQ]]-1 (* _Harvey P. Dale, Nov 29 2017 *)

A255532 Indices of primes in the 9th-order Fibonacci number sequence, A251749.

Original entry on oeis.org

10, 14, 19, 29, 404, 1744, 8854, 27754

Offset: 1

Robert Price, Feb 24 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, A251749.

a={0,0,0,0,1,0,0,0,0}; step=9; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst

A255533 Indices of primes in the 9th-order Fibonacci number sequence, A251750.

Original entry on oeis.org

10, 33, 43, 253, 1253, 2389

Offset: 1

Robert Price, Feb 24 2015

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, A000383, A249413, A060455, A079262, A251750.

a={0,0,0,1,0,0,0,0,0}; step=9; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst

A255534 Indices of primes in the 9th-order Fibonacci number sequence, A251751.

Original entry on oeis.org

10, 12, 232, 502

Offset: 1

Robert Price, Feb 24 2015

a(5) > 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, A251751.

a={0,0,1,0,0,0,0,0,0}; step=9; 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[Table[1,{9}],{0,0,1,0,0,0,0,0,0},510], ?(PrimeQ[#]&)]]-1 (* _Harvey P. Dale, Feb 27 2016 *)

A255536 Indices of primes in the 9th-order Fibonacci number sequence, A251752.

Original entry on oeis.org

10, 11, 21, 29, 301, 57089

Offset: 1

Robert Price, Feb 24 2015

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, A000383, A249413, A060455, A079262, A251752.

a={0,1,0,0,0,0,0,0,0}; step=9; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst

A345669 Antidiagonal sums of array containing i-bonacci sequences nac(i,n), where nac(i,n) is the n-th i-bonacci number with nac(i,1..i) = 1 (see comments).

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 18, 31, 51, 89, 153, 273, 483, 870, 1571, 2860, 5225, 9603, 17711, 32805, 60967, 113685, 212610, 398723, 749615, 1412585, 2667549, 5047345, 9567527, 18166272, 34546857, 65793343, 125471295, 239584610, 458028439, 876628109, 1679581899

Offset: 1

Christoph B. Kassir, Jun 21 2021

nonn

Antidiagonal sum of below array:
1, 1, 1, 1, 1, 1, ... (1-bonacci numbers)
1, 1, 2, 3, 5, 8, ... (2-bonacci or Fibonacci numbers)
1, 1, 1, 3, 5, 9, ... (3-bonacci or tribonacci numbers)
1, 1, 1, 1, 4, 7, ... (4-bonacci or tetranacci numbers)
...

Cf. A000045, A000213, A000288, A000322, A000383, A060455, A061451.

b:= proc(i, n) option remember; `if`(n=0, 0,
      `if`(n<=i, 1, add(b(i, n-j), j=1..i)))
    end:
a:= n-> add(b(i+1, n-i), i=0..n):
seq(a(n), n=1..37);  # Alois P. Heinz, Jun 21 2021

b[i_, n_] := b[i, n] = If[n == 0, 0, If[n <= i, 1, Sum[b[i, n - j], {j, 1, i}]]];
a[n_] := Sum[b[i + 1, n - i], {i, 0, n}];
Table[a[n], {n, 1, 37}] (* Jean-François Alcover, Dec 27 2022, after Alois P. Heinz *)

a(n) = Sum_{i=1..n} of nac(i,n-i+1) = Sum_{i=1..n} of nac(n-i+1,i).

cp's OEIS Frontend

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

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

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A255529 Indices of primes in the 9th-order Fibonacci number sequence, A104144.

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A255530 Indices of primes in the 9th-order Fibonacci number sequence, A251746.

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A255531 Indices of primes in the 9th-order Fibonacci number sequence, A251747.

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A255532 Indices of primes in the 9th-order Fibonacci number sequence, A251749.

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A255533 Indices of primes in the 9th-order Fibonacci number sequence, A251750.

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A255534 Indices of primes in the 9th-order Fibonacci number sequence, A251751.

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A255536 Indices of primes in the 9th-order Fibonacci number sequence, A251752.