cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A178762 Prime numbers that are Fibonacci integers.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 61, 89, 107, 199, 211, 233, 281, 421, 521, 1103, 1597, 2161, 2207, 2521, 3001, 3571, 5779, 9349, 9901, 14503, 19801, 28657, 90481, 103681, 109441, 135721, 141961, 514229, 3010349, 6376021, 11128427
Offset: 1

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Author

T. D. Noe, Jun 10 2010

Keywords

Comments

A Fibonacci integer is a number that can be written as the product and/or quotient of Fibonacci numbers (A000045). For example, 107 is a Fibonacci integer because Fib(36)/(Fib(18)*Fib(3)*Fib(4)^3) = 107. Observe that the prime Fibonacci numbers (A005478) are a subset of these primes. Luca, Pomerance, and Wagner conjecture that this sequence is infinite. The paper's Remark 2 and sequences A152012, A178763, and A178764 are useful in finding these primes.

Links

A178764 Ratio of the primitive part of Fibonacci(n) to the product of primitive prime factors of Fibonacci(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

T. D. Noe, Jun 10 2010

Keywords

Comments

Except for a(6), it can be shown that all terms are either 1 or prime. In the first 10^5 terms, only 151 are greater than 1.

Links

Formula

a(n) = A061446(n) / A178763(n).

A308949 a(n) is the greatest divisor of A000129(n) that is coprime to A000129(m) for all positive integers m < n.

Original entry on oeis.org

1, 2, 5, 3, 29, 7, 169, 17, 197, 41, 5741, 11, 33461, 239, 269, 577, 1136689, 199, 6625109, 1121, 45697, 8119, 225058681, 1153, 45232349, 47321, 7761797, 38081, 44560482149, 961, 259717522849, 665857, 52734529, 1607521, 1800193921, 13067, 51422757785981
Offset: 1

Views

Author

Jianing Song, Jul 02 2019

Keywords

Comments

a(n) is squarefree unless n is of the form A214028(A238736(k)) = {7, 30, 1546462, ...}. The terms in A238736 are called 2-Wall-Sun-Sun primes.

Examples

			A000129(30) = 107578520350 = 2 * 5^2 * 7 * 29 * 31^2 * 41 * 269. We have 2, 7 divides A000129(6) = 70; 29, 41 divides A000129(10) = 2378; 5, 269 divides A000129(15) = 195025, but A000129(m) is coprime to 31 for all 1 <= m < 30, so a(30) = 31^2 = 961.

Crossrefs

Cf. A000129, A008555, A178763, A214028, A238736.

Programs

  • Mathematica
    nmax = 40;
pell = {1, 2};
pp = {1, 2};
Do[s = 2*pell[[-1]] + pell[[-2]];
  AppendTo[pell, s];
  AppendTo[pp, s/Times @@ pp[[Most[Divisors[n]]]]], {n, 3, nmax}];
a[2] = 2;
a[n_] := pp[[n]]/GCD[pp[[n]], n];
Array[a, nmax] (* Jean-François Alcover, Jul 06 2019, after T. D. Noe in A008555 *)
  • PARI
    T(n) = ([2, 1; 1, 0]^n)[2, 1]
b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
a(n) = if(n==2, 2, b(n)/gcd(n, b(n)))

Formula

a(n) = A008555(n) / gcd(A008555(n), n) if n != 2.

A309525 a(n) is the greatest divisor of A006190(n) that is coprime to A006190(m) for all positive integers m < n.

Original entry on oeis.org

1, 3, 10, 11, 109, 1, 1189, 119, 1297, 131, 141481, 59, 1543321, 1429, 3089, 14159, 183642229, 433, 2003229469, 14041, 1837837, 170039, 238367471761, 7079, 23854956949, 1854841, 2186871697, 1670761, 309400794703549, 12871, 3375045015828949, 200477279
Offset: 1

Views

Author

Jianing Song, Aug 06 2019

Keywords

Comments

Analog of A178763 and A308949.

Examples

			A006190(12) = 467280 = 2^4 * 3^2 * 5 * 11 * 59. We have 2, 3, 5 divides A006190(6) = 360 and 11 divides A006190(3) = 11, but A006190(m) is coprime to 59 for all 1 <= m < 12, so a(12) = 59.

Links

Crossrefs

Cf. A006190, A253807.
Cf. A178763, A308949, A309526.

Programs

  • Maple
    A6190:= proc(n) option remember; 3*procname(n-1)+procname(n-2) end proc:
A6190(0):= 0: A6190(1):= 1:
f:= proc(n) local k,i,g;
  k:= A6190(n);
  for i from 1 to n-1 do
    g:= igcd(k,A6190(i));
    while g > 1 do
      k:= k/g;
      g:= igcd(k,A6190(i));
    od;
  od;
  k
end proc:
map(f, [$1..40]); # Robert Israel, Aug 02 2024
  • PARI
    T(n) = ([3, 1; 1, 0]^n)[2, 1]
b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
a(n) = if(isprime(n)&&!(13%n), 1543321, if(n!=6, b(n)/gcd(n, b(n)), 1))

Formula

a(n) = A253807(n) / gcd(A253807(n), n) if n != 6, 13.

A309526 a(n) is the greatest divisor of A001353(n) that is coprime to A001353(m) for all positive integers m < n.

Original entry on oeis.org

1, 4, 15, 7, 209, 13, 2911, 97, 901, 181, 564719, 193, 7865521, 2521, 6989, 18817, 1525870529, 2701, 21252634831, 37441, 6779137, 489061, 4122901604639, 37633, 274758906449, 6811741, 6575588101, 1037623, 11140078609864049, 40321, 155161278879431551
Offset: 1

Views

Author

Jianing Song, Aug 06 2019

Keywords

Comments

Analog of A178763 and A308949.
Let b(n) = A309040(n)*gcd(A309040(n),n), then for n > 3: a(n) = b(2n) for even n and b(n)*b(2n) for odd n. It seems highly impossible that b(n) = 1 holds for n > 3, so it seems that only even-indexed terms can be primes.

Examples

			A001353(6) = 780 = 2^2 * 3 * 5 * 13. We have 2 divides A001353(2) = 2 and 3, 5 divides A001353(3) = 15, but A001353(m) is coprime to 13 for all 1 <= m < 6, so a(6) = 13.

Crossrefs

Cf. A001353, A306825, A309040.
Cf. A178763, A308949, A309525.

Programs

  • PARI
    T(n) = ([4, -1; 1, 0]^n)[2, 1]
b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
a(n) = if(isprime(n)&&!(12%n), b(n), b(n)/gcd(n, b(n)))

Formula

a(n) = A306825(n) / gcd(A306825(n), n) if n != 2, 3.
Showing 1-5 of 5 results.

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