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.

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A278637 Numbers k such that Fibonacci(k) is either prime or semiprime.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 94, 101, 107, 109, 113, 121, 127, 131, 137, 151, 167, 173, 191, 193, 199, 227, 251, 271, 277, 293, 331, 353, 359, 397, 401, 431, 433, 449, 467, 509, 569, 571, 587, 599, 601, 613, 631, 653, 743, 991, 1091, 1223, 1373, 1487
Offset: 1

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Author

Bobby Jacobs, Jan 04 2017

Keywords

Comments

Numbers k such that 2 <= A063375(k) <= 4.
All numbers in the first 4 rows of A279021 are in this sequence (3, 5, 11, 17, 353, 431, 509, and 587).
Are all numbers of A279021 in this sequence?

Crossrefs

Cf. A000045, A001605, A063375, A072381, A136341, A279021.

Formula

A001605 U A072381.

Extensions

a(31)-a(64) from Charles R Greathouse IV, Jan 04 2017
a(65)-a(73) from Max Alekseyev, Feb 26 2023

A346491 Number of factorizations of the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 2, 2, 2, 1, 29, 1, 2, 5, 5, 1, 21, 2, 15, 5, 2, 1, 719, 4, 2, 15, 15, 1, 296, 2, 15, 5, 2, 5, 4323, 5, 5, 5, 203, 2, 296, 1, 52, 52, 5, 1, 32653, 5, 135, 5, 15, 2, 1315, 15, 566, 52, 5, 2, 270920, 2, 5, 52, 203, 5, 296, 5, 52, 52, 877, 2
Offset: 1

Views

Author

Alois P. Heinz, Jul 19 2021

Keywords

Links

Crossrefs

Cf. A000045, A001055, A001605, A046523, A072381, A278245.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n>k, 0, 1)+`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=numtheory[divisors](n) minus {1, n}))
    end:
a:= proc(n) option remember; b((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
      sort(map(i-> i[2], ifactors(combinat[fibonacci](n))[2]), `>`))$2)
    end:
seq(a(n), n=1..80);
  • Mathematica
    T[, 1] = T[1, ] = 1;
T[n_, m_] := T[n, m] = DivisorSum[n, If[1 < # <= m, T[n/#, #], 0]&];
f[n_] := T[n, n];
a[n_] := f[Fibonacci[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 119}] (* Jean-François Alcover, Sep 08 2022 *)

Formula

a(n) = A001055(A000045).
a(n) = A001055(A046523(A000045(n))).
a(n) = A001055(A278245(n)).
a(n) = 1 <=> n in { A001605 } union {1,2}.
a(n) = 2 <=> n in { A072381 }.

A363837 Numbers k such that k-th Jacobsthal number A001045(k) is a semiprime.

Original entry on oeis.org

6, 8, 10, 14, 26, 29, 34, 37, 38, 41, 47, 49, 53, 62, 67, 71, 73, 103, 107, 109, 122, 139, 151, 179, 223, 229, 251, 254, 269, 277, 311, 349, 353, 433, 457, 487, 503, 599, 601, 613, 619, 643, 739, 757, 827, 839, 1031, 1061, 1117
Offset: 1

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Author

Sean A. Irvine, Oct 19 2023

Keywords

Examples

			10 is a term because Jacobsthal(10) = A001045(10) = 341 = 11*31 is a semiprime.

Links

Crossrefs

Cf. A001045, A001358, A277356 (the actual semiprimes), A250292, A085726, A072381, A101757, A286567, A271314.

Programs

  • PARI
    isok(k) = bigomega((2^k - (-1)^k)/3) == 2; \\ Michel Marcus, Oct 19 2023

Extensions

a(47)-a(49) from Amiram Eldar, Feb 25 2024

A206097 Fibonacci numbers F that are squarefree semiprimes such that F+2 or F-2 is also a squarefree semiprime.

Original entry on oeis.org

55, 4181, 17711, 121393, 5702887
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 03 2012

Keywords

Comments

a(6) > Fibonacci(1500), if it exists. - Amiram Eldar, Aug 01 2024

Examples

			55 = Fibonacci(10) is a term because 55 = 5 * 11 and 55 + 2 = 57 = 3 * 19 are both squarefree semiprimes.
4181 = Fibonacci(19) is a term because 4181 = 37 * 113 and 4181 + 2 = 4183 = 47 * 89 are both squarefree semiprimes.

Crossrefs

Subsequence of A000045, A006881 and A053409.
Cf. A072381.

Programs

  • Mathematica
    Select[Fibonacci[Range[300]], Last/@FactorInteger[#]=={1,1} && (Last/@FactorInteger[#+2]=={1,1} || Last/@FactorInteger[#-2]=={1,1})&]

Extensions

Name corrected by Amiram Eldar, Aug 01 2024
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