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 1-4 of 4 results.

A072381 Numbers m such that Fibonacci(m) is a semiprime.

Original entry on oeis.org

8, 9, 10, 14, 19, 22, 26, 31, 34, 41, 53, 59, 61, 71, 73, 79, 89, 94, 101, 107, 109, 113, 121, 127, 151, 167, 173, 191, 193, 199, 227, 251, 271, 277, 293, 331, 353, 397, 401, 467, 587, 599, 601, 613, 631, 653, 743, 991, 1091, 1223, 1373, 1487
Offset: 1

Views

Author

Shyam Sunder Gupta, Jul 20 2002

Keywords

Comments

Note that there are two cases: (1) n is 2p, in which case the semiprime is Fibonacci(p)*Lucas(p) for some prime p, or (2) n is a power of a prime p^k for k > 0. In the first case, the primes p are in sequence A080327. In the second case, it appears that k=1 except for n = 8, 9 and 121. - T. D. Noe, Sep 23 2005
The associated sequence of Fibonacci numbers contains no squares, since the only Fibonacci numbers which are square are 1 and 144. Consequently this is a subsequence of A114842. - Charles R Greathouse IV, Sep 24 2012
Sequence continues as 1543?, 1709, 1741?, 1759, 1801?, 1889, 1987, ..., where ? marks uncertain terms. - Max Alekseyev, Jul 10 2016

Examples

			a(4) = 14 because the 14th Fibonacci number 377 = 13*29 is a semiprime.

Links

Crossrefs

Cf. A053409, A085726 (n such that n-th Lucas number is a semiprime).
Column k=2 of A303215.

Programs

  • Mathematica
    Select[Range[200], Plus@@Last/@FactorInteger[Fibonacci[ # ]] == 2&] (Noe)
Select[Range[1500],PrimeOmega[Fibonacci[#]]==2&] (* Harvey P. Dale, Dec 13 2020 *)
  • PARI
    for(n=2,9999,bigomega(fibonacci(n))==2&&print1(n",")) \\ - M. F. Hasler, Oct 31 2012
  • PARI
    issemi(n)=bigomega(n)==2
is(n)=if(n%2, my(p); if(issquare(n,&p), isprime(p) && isprime(fibonacci(p)) && isprime(fibonacci(n)/fibonacci(p)), isprime(n) && issemi(fibonacci(n))), (isprime(n/2) && isprime(fibonacci(n/2)) && isprime(fibonacci(n)/fibonacci(n/2))) || n==8) \\ Charles R Greathouse IV, Oct 06 2016

Extensions

More terms from Don Reble, Jul 31 2002
a(49)-a(50) from Max Alekseyev, Aug 18 2013
a(51)-a(52) from Max Alekseyev, Jul 10 2016

A250292 Numbers k such that Pell(k) is a semiprime.

Original entry on oeis.org

7, 9, 17, 19, 23, 43, 47, 67, 73, 83, 103, 109, 139, 149, 157, 173, 179, 223, 239, 281, 311, 313, 349, 431, 557, 569, 577, 587
Offset: 1

Views

Author

Eric Chen, Dec 24 2014

Keywords

Comments

a(29) >= 709. - Hugo Pfoertner, Jul 29 2019
859, 937, 1087, 1151, and 1193 belong to the sequence. 709 and 787 have not yet been ruled out. The next candidate after these appears to be 1471. - Daniel M. Jensen, Oct 18 2019

Examples

			17 is a term since Pell(17) = 1136689 = 137 * 8297 is a semiprime.

Links

Crossrefs

Cf. A001358, A096650, A072381, A085726, A000129.

Programs

  • Maple
    pell:= gfun:-rectoproc({a(0)=0,a(1)=1,a(n)=2*a(n-1)+a(n-2)},a(n),remember):
filter:= proc(n) local F,f;
   F:= ifactors(pell(n),easy)[2];
   if add(f[2],f=F) > 2 then return false fi;
   if hastype(F,symbol) then
     if add(f[2],f=F) >= 2 then return false fi;
   else return evalb(add(f[2],f=F)=2)
   fi;
   F:= ifactors(pell(n))[2];
   evalb(add(f[2],f=F)=2)
end proc:
select(filter, [$1..230]); # Robert Israel, Jan 18 2016
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + a[n - 2]; Select[Range[0, 160], PrimeOmega@ a@ # == 2 &] (* Michael De Vlieger, Jan 19 2016 *)

Extensions

a(22)-a(23) from Daniel M. Jensen, Jan 18 2016
a(24) from Arkadiusz Wesolowski, Jan 19 2016
a(25)-a(27) from Sean A. Irvine, Jul 17 2017
a(28) from Sean A. Irvine, Jan 24 2018

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

Views

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

A129745 Numbers k such that Lucas(4k)/7 is prime.

Original entry on oeis.org

5, 17, 19, 41, 43, 71, 1511, 2339, 3469, 4787, 7211, 9781, 14431
Offset: 1

Views

Author

Alexander Adamchuk, May 14 2007, May 16 2007

Keywords

Comments

L(m) = Lucas(m) = Fibonacci(m-1) + Fibonacci(m+1). 7 = L(4) divides L(4*(1+2m)) since L(4m) = L(4)*L(4*(m-1)) - L(4*(m-2)).
Integer k is in this sequence iff k is prime and 4k belongs to A085726. - Max Alekseyev, May 16 2010
a(14) > 60000. - Michael S. Branicky, Aug 01 2024

Crossrefs

Cf. A000032, A001606 (indices of prime Lucas numbers).
Cf. A074304 (numbers k such that Lucas(2k)/3 is prime).

Programs

  • Mathematica
    a=7; b=47; Do[ c=7b-a; a=b; b=c; If[ PrimeQ[c/7], Print[n] ], {n, 3, 100}]

Extensions

a(7) - a(10) from Stefan Steinerberger, May 17 2007
a(11) from Max Alekseyev, Nov 25 2007
a(12) from Alexander Adamchuk, May 15 2010
a(13) from Michael S. Branicky, Aug 01 2024
Showing 1-4 of 4 results.

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