A065069 -id:A065069 - cp's OEIS frontend

A237043 Numbers n such that 2^n - 1 is not squarefree, but 2^d - 1 is squarefree for every proper divisor d of n.

6, 20, 21, 110, 136, 155, 253, 364, 602, 657, 812, 889, 979, 1081

Offset: 1

Charles R Greathouse IV, Feb 02 2014

Primitive elements of A049094: the elements of A049094 are precisely the positive multiples of members of this sequence.
If p^2 divides 2^n - 1 for some odd prime p, then by definition the multiplicative order of 2 mod p^2 divides n. The multiplicative order of 2 mod p^2 is p times the multiplicative order of 2 mod p unless p is a Wieferich prime, in which case the two orders are identical. Hence either p is a Wieferich prime or p*log_2(p+1) <= n. This should allow finding larger members of this sequence. - Charles R Greathouse IV, Feb 04 2014
If n is in the sequence and m>1 then m*n is not in the sequence. Because n is a proper divisor of m*n and 2^n-1 is not squarefree. - Farideh Firoozbakht, Feb 11 2014
a(15) >= 1207. - Max Alekseyev, Sep 28 2015
From Daniel Suteu, Jul 03 2019: (Start)
The following numbers are also in the sequence: {1755, 2265, 2485, 2756, 3081, 3164, 4112, 6757, 8251, 13861, 18533}.
Probably, the following numbers are also terms: {3422, 5253, 6806, 8164, 9316, 11342, 12550, 15025, 15026, 17030, 17404, 17468, 18145, 19670, 19701, 22052}. (End)

Peice Hua, Finite 2-groups having a cyclic or dihedral maximal subgroup and arc-transitive maps, arXiv:2508.05981 [math.GR], 2025. See p. 3.

Cf. A049094, A005420, A065069, A282631, A282632.

Select[Range@ 160, And[AllTrue[2^#2 - 1, SquareFreeQ], ! SquareFreeQ[2^First@ #1 - 1]] & @@ TakeDrop[Divisors@ #, -1] &] (* Michael De Vlieger, Jul 07 2019 *)

default(factor_add_primes, 1);
isA049094(n)=my(f=factor(n>>valuation(n, 2))[, 1], N, o); for(i=1, #f, if(n%(f[i]-1) == 0, return(1))); N=2^n-1; fordiv(n, d, f=factor(2^d-1)[, 1]; for(i=1, #f, if(d==n, return(!issquarefree(N))); o=valuation(N, f[i]); if(o>1, return(1)); N/=f[i]^o))
is(n)=fordiv(n,d,if(isA049094(d),return(d==n))); 0

\\ Simpler but slow
is(n)=fordiv(n,d,if(!issquarefree(2^d-1),return(d==n))); 0

a(14) from Charles R Greathouse IV, Sep 21 2015, following Womack's factorization of 2^991-1.

A037917 Numbers n such that the Fibonacci number F(n) is divisible by a square.

Original entry on oeis.org

6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, 72, 75, 78, 84, 90, 91, 96, 100, 102, 108, 110, 112, 114, 120, 125, 126, 132, 138, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 225, 228, 234, 240

Offset: 1

Marc LeBrun

nonn

Is a(n) asymptotic to C*n with 4 < C < 4.5 ? - Benoit Cloitre, Sep 04 2002
Numbers are a superset of the multiples of 6 (A008588), because 8 divides Fibonacci(6m) = A134492(m). Sequence apparently also contains the multiples of 25. Are all a(n) composite? Members not divisible by 6 or 25 are 56, 91, 110, 112, 153, 182, 220, 224, 273, 280, ... - Ralf Stephan, Jan 26 2014
These numbers are the positive multiples of A065069. - Charles R Greathouse IV, Feb 02 2014
To address Cloitre's question, if such C exists it must be less than 4.3 using the known terms of A065069. - Charles R Greathouse IV, Feb 04 2014

Amiram Eldar, Table of n, a(n) for n = 1..328 (terms 1..235 from T. D. Noe, based on Kelly's table)
Blair Kelly, Fibonacci Factorizations
Eric Weisstein's World of Mathematics, Fibonacci Number.

Cf. A000045, A037918, A001177, A030426.

Select[Range[100], MoebiusMu[Fibonacci[#]] == 0 &] (* Alonso del Arte, Jan 26 2014 *)

is(n)=!issquarefree(fibonacci(n)) \\ Charles R Greathouse IV, Feb 02 2014

More terms from Eric W. Weisstein

A065106 Smallest Fibonacci index to produce a factor p^2 (for primes p).

Original entry on oeis.org

6, 12, 25, 56, 91, 110, 153, 342, 406, 552, 703, 752, 820, 915, 930, 979

Offset: 1

Len Smiley, Nov 21 2001

nonn

Following Lucas, these might be called the prime-squared ranks of apparition.

Assuming that there are no square primitive factors in the Fibonacci sequence (an open question), then this sequence continues 1431, 1892, 2147, 2701, 2943, 3029, 3422, 3852, 4378, 4556, 4753, 4970, 5050, 5513, 6162, 6394, 6972, 7550, 7868, 8841, 8862, 9453. This is obtained by sorting the sequence prime(n)*A001602(n). - T. D. Noe, Apr 15 2004

			342 is here but not in A065069 because Fib(342) is the first Fib divisible by 19^2, but 342 is divisible by 6 and so is not a primitive index.

Cf. A065069, A037917, A065107.

Cf. A001602 (smallest m such that prime(n) divides Fibonacci(m)).

A065107 Start of the permutation of the primes in the order in which p^2 first appears as a factor of a number in the Fibonacci sequence.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 17, 19, 29, 23, 37, 47, 41, 61, 31, 89

Offset: 1

Len Smiley, Nov 21 2001

Assuming that there are no square primitive factors in the Fibonacci sequence (an open question), then this sequence continues 53, 43, 113, 73, 109, 233, 59, 107, 199, 67, 97, 71, 101, 149, 79, 139, 83, 151, 281, 421, 211, 137, 103, 157, 307, 521. This is obtained by sorting the pairs (prime(n)*A001602(n), prime(n)) by the first position and noting the order of the primes in the second position. - T. D. Noe, Apr 15 2004

Cf. A065069, A065106.

Cf. A001602 (smallest m such that prime(n) divides Fibonacci(m)).

A264008 Index of the smallest Fibonacci number divisible by prime(n)^2.

Original entry on oeis.org

6, 12, 25, 56, 110, 91, 153, 342, 552, 406, 930, 703, 820, 1892, 752, 1431, 3422, 915, 4556, 4970, 2701, 6162, 6972, 979, 4753, 5050, 10712, 3852, 2943, 2147, 16256, 17030, 9453, 6394, 5513, 7550, 12403, 26732, 28056, 15051, 31862, 16290, 36290, 18721, 19503, 4378, 8862, 49952, 51756, 26106, 3029, 56882, 28920

Offset: 1

R. J. Mathar, Oct 31 2015

Robert Israel, Table of n, a(n) for n = 1..3584

Cf. A065069, A065106.

f:= proc(n) local p, phi,q,k,G,Fkm,Fk,M,W,m;
  p:= ithprime(n);
  if member(p mod 5, {1,4}) then
    phi:= rhs(op(msolve(x^2-x-1,p^2)[1]));
    q:= -1-phi mod p^2;
    return numtheory:-order(q,p^2);
  fi;
  G:= GF(p,2,alpha^2-alpha-1);
  q:= G:-ConvertIn(-1-alpha);
  k:= G:-order(q);
  Fkm:= combinat:-fibonacci(k-1) mod p^2;
  Fk:= combinat:-fibonacci(k) mod p^2;
  M:= <|>;
  W:= <0,1>;
  for m from 1 do
     W:= M . W mod p^2;
     if W[1] = 0 then return(m*k) fi
  od:
end proc:
f(3):= 25:
map(f, [$1..100]); # Robert Israel, Jan 04 2018

a(n) = if(n==3, 25, my(p=prime(n)); fordiv(p^2-1, d, if(fibonacci(d)%p==0, return(d*p)))); \\ Altug Alkan, Oct 31 2015

a(n) = prime(n)*A001602(n).

a(n) = min{i: A001248(n) | A000045(i)}

cp's OEIS Frontend

A237043 Numbers n such that 2^n - 1 is not squarefree, but 2^d - 1 is squarefree for every proper divisor d of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A037917 Numbers n such that the Fibonacci number F(n) is divisible by a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A065106 Smallest Fibonacci index to produce a factor p^2 (for primes p).

Views

Author

Keywords

Comments

Examples

Crossrefs

A065107 Start of the permutation of the primes in the order in which p^2 first appears as a factor of a number in the Fibonacci sequence.

Views

Author

Keywords

Comments

Crossrefs

A264008 Index of the smallest Fibonacci number divisible by prime(n)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula