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.

A140409 Prime factors of Lucas numbers.

Original entry on oeis.org

2, 3, 7, 11, 19, 23, 29, 31, 41, 43, 47, 59, 67, 71, 79, 83, 101, 103, 107, 127, 131, 139, 151, 163, 167, 179, 181, 191, 199, 211, 223, 227, 229, 239, 241, 251, 263, 271, 281, 283, 307, 311, 331, 347, 349, 359, 367, 379, 383, 401, 409, 419, 431, 439, 443, 449
Offset: 1

Views

Author

Tanya Khovanova, Jun 16 2008

Keywords

Comments

a(n) is A096362(n) in ascending order.

Links

Crossrefs

Cf. A096362 Order in which prime factors first occur in the Lucas sequence.

Formula

Union of 2, A053027 and A053032 - T. D. Noe, Jun 21 2008

Extensions

More terms from T. D. Noe, Jun 21 2008

A160243 a(n) = Lucas(n+1) + prime(n).

Original entry on oeis.org

5, 7, 12, 18, 29, 42, 64, 95, 146, 228, 353, 558, 884, 1407, 2254, 3624, 5837, 9410, 15194, 24547, 39676, 64158, 103765, 167850, 271540, 439305, 710750, 1149958, 1860607, 3010462, 4870974, 7881327, 12752180, 20633378, 33385431, 54018672, 87403960, 141422487
Offset: 1

Views

Author

Enoch Haga, May 05 2009

Keywords

Comments

Lucas(n) = A000032(n), prime(n) = A000040(n).

Examples

			a(1) = Lucas(2) + prime(1) = 3 + 2 = 5.
a(4) = Lucas(5) + prime(4) = 11 + 7 = 18.

Links

Crossrefs

A000032 (Lucas numbers, beginning at 2), A000040 (primes), A096362 (order in which prime factors first occur in the Lucas sequence), A160244 (A000285(n) + A000040(n)).

Programs

  • Magma
    [ Lucas(n+1)+NthPrime(n): n in [1..40] ]; // Klaus Brockhaus, May 20 2009
  • Mathematica
    Table[LucasL[n+1]+Prime[n],{n,40}] (* Harvey P. Dale, May 25 2021 *)
  • UBASIC
    10 'Lucas variations (change value of A in line 30 as appropriate) 20 P=1 30 A=2 40 B=1 50 C=A+B:print C;:R=nxtprm(P):print R;:P=R:print C+R 51 if C=prmdiv(C) then print C;"*":U=U+1 52 if C+R=prmdiv(C+R) then print C+R;"#":V=V+1 60 D=B+C:print D;:R=nxtprm(P):print R;:P=R:print D+R 61 if D=prmdiv(D) then print D;"*":U=U+1 62 if D+R=prmdiv(D+R) then print D+R;"#":V=V+1 63 print U;V 70 stop 80 A=C:B=D:goto 50

Extensions

Edited by Klaus Brockhaus, May 20 2009
Corrected and extended by Harvey P. Dale, May 25 2021

A121708 Numerator of Sum/Product of first n Fibonacci numbers A000045[n].

Original entry on oeis.org

1, 2, 2, 7, 2, 1, 11, 3, 11, 1, 29, 47, 29, 1, 19, 41, 19, 1, 199, 23, 199, 1, 521, 281, 521, 1, 31, 2207, 31, 1, 3571, 107, 3571, 1, 9349, 2161, 9349, 1, 211, 13201, 211, 1, 64079, 1103, 64079, 1, 15251, 90481, 15251, 1, 5779, 14503, 5779, 1, 1149851, 2521
Offset: 1

Views

Author

Alexander Adamchuk, Aug 16 2006, Sep 21 2006

Keywords

Comments

a(1) = 1 and a(4k+2) = 1 for k>0.
For k >1 a(4k-1) = a(4k+1) = A072183(2k+1) = A061447(2k+1) Primitive part of Lucas(n).

Crossrefs

Cf. A000045, A000032, A079451, A001578, A096362, A058036, A121709, A090585.
Cf. A061447, A072183.

Programs

  • Mathematica
    Table[Numerator[Sum[Fibonacci[k],{k,1,n}]/Product[Fibonacci[k],{k,1,n}]],{n,1,100}]
With[{fibs=Fibonacci[Range[60]]},Numerator[Accumulate[fibs]/Rest[ FoldList[ Times,1,fibs]]]] (* This is significantly faster than the first program above *) (* Harvey P. Dale, Aug 19 2012 *)

Formula

a(n) = numerator( sum(k=1..n, Fibonacci(k)) / prod(k=1..n, Fibonacci(k)) ).

A121709 Numerator of Sum/Product of first n Lucas numbers A000032[n].

Original entry on oeis.org

1, 4, 2, 5, 13, 1, 73, 5, 7, 1, 37, 5, 1361, 1, 223, 25, 4673, 1, 24473, 25, 16019, 1, 83879, 65, 62743, 1, 20533, 65, 1505173, 1, 7881193, 85, 5158309, 1, 27009259, 425, 1400221, 1, 1446283, 2225, 69237359, 1, 51790217, 445, 1660959719, 1, 8696897999
Offset: 1

Views

Author

Alexander Adamchuk, Aug 16 2006

Keywords

Comments

5 divides a(4k). a(1) = 1 and a(4k+2) = 1 for k>0.

Crossrefs

Cf. A000045, A000032, A079451, A001578, A096362, A058036, A121708, A090585.

Programs

  • Mathematica
    Table[Numerator[Sum[Fibonacci[k-1]+Fibonacci[k+1],{k,1,n}]/Product[Fibonacci[k-1]+Fibonacci[k+1],{k,1,n}]],{n,1,100}]

Formula

a(n) = Numerator[Sum[Lucas[k],{k,1,n}]/Product[Lucas[k],{k,1,n}]], where Lucas[k] = Fibonacci[k-1] + Fibonacci[k+1].

A160244 A104449(n+1)+prime(n), sum of a Lucas and the prime sequence.

Original entry on oeis.org

6, 8, 14, 21, 34, 50, 77, 116, 180, 283, 442, 702, 1117, 1784, 2864, 4611, 7434, 11994, 19375, 31312, 50622, 81869, 132422, 214218, 346565, 560698, 907168, 1467769, 2374836, 3842502, 6217243, 10059636, 16276758, 26336265, 42612896, 68949024
Offset: 1

Views

Author

Enoch Haga, May 05 2009

Keywords

Comments

A104449 could be called L31(n), the Lucas sequence starting with 3,1.

Crossrefs

Cf. A000285, A096362, A160243.

Programs

  • UBASIC
    10 'Lucas variations (change value of A in line 30 as appropriate) 20 P=1 30 A=2 40 B=1 50 C=A+B:print C;:R=nxtprm(P):print R;:P=R:print C+R 51 if C=prmdiv(C) then print C;"*":U=U+1 52 if C+R=prmdiv(C+R) then print C+R;"#":V=V+1 60 D=B+C:print D;:R=nxtprm(P):print R;:P=R:print D+R 61 if D=prmdiv(D) then print D;"*":U=U+1 62 if D+R=prmdiv(D+R) then print D+R;"#":V=V+1 63 print U;V 70 stop 80 A=C:B=D:goto 50

Formula

a(n) = A000285(n)+A000040(n).

Extensions

Edited by R. J. Mathar, May 12 2009
Showing 1-5 of 5 results.

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