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 1-6 of 6 results.

A086927 a(n) = 10*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 10.

Original entry on oeis.org

2, 10, 102, 1030, 10402, 105050, 1060902, 10714070, 108201602, 1092730090, 11035502502, 111447755110, 1125513053602, 11366578291130, 114791295964902, 1159279537940150, 11707586675366402, 118235146291604170
Offset: 0

Views

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 21 2003

Keywords

Comments

a(n+1)/a(n) converges to (5+sqrt(26)) = 10.099019...
Lim a(n)/a(n+1) as n approaches infinity = 0.099019... = 1/(5+sqrt(26)) = (sqrt(26)-5).

Examples

			a(4) = 10402 = 10*a(3) + a(2) = 10*1030 + 102 = (5+sqrt(26))^4 + (5-sqrt(26))^4 =  10401.999903 + 0.000097 = 10402.

References

  • Stefano Arnone, C Falcolini, F Moauro, M Siccardi, On Numbers in Different Bases: Symmetries and a Conjecture, Experimental Mathematics, Vol 26 2016, pp 197-209; http://dx.doi.org/10.1080/10586458.2016.1149125

Links

Crossrefs

Cf. A036336.

Programs

  • Magma
    I:=[2,10]; [n le 2 select I[n] else 10*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016
  • Mathematica
    RecurrenceTable[{a[0] == 2, a[1] == 10, a[n] == 10 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)

Formula

a(n) = (5+sqrt(26))^n + (5-sqrt(26))^n.
G.f.: (2-10*x)/(1-10*x-x^2). - Philippe Deléham, Nov 20 2008
a(n) = 2*A088320(n). - R. J. Mathar, Feb 06 2020

Extensions

More terms from Jon E. Schoenfield, May 15 2010

A087799 a(n) = 10*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 10.

Original entry on oeis.org

2, 10, 98, 970, 9602, 95050, 940898, 9313930, 92198402, 912670090, 9034502498, 89432354890, 885289046402, 8763458109130, 86749292044898, 858729462339850, 8500545331353602, 84146723851196170, 832966693180608098, 8245520207954884810
Offset: 0

Views

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 11 2003

Keywords

Comments

a(n+1)/a(n) converges to (5+sqrt(24)) = 9.8989794... a(0)/a(1)=2/10; a(1)/a(2)=10/98; a(2)/a(3)=98/970; a(3)/a(4)=970/9602; ... etc. Lim a(n)/a(n+1) as n approaches infinity = 0.10102051... = 1/(5+sqrt(24)) = (5-sqrt(24)).
Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 96 = 0. - Colin Barker, Feb 25 2014
A triangle whose sides are a(n) - 1, a(n) and a(n) + 1 is nearly Fleenor-Heronian since its area is the product of an integer and the square root of 2. See A003500. - Charlie Marion, Dec 18 2020

Examples

			a(4) = 9602 = 10*a(3) - a(2) = 10*970 - 98 = (5+sqrt(24))^4 + (5-sqrt(24))^4.

Links

Crossrefs

Cf. A005248, A006242, A036336, A086927, A135927, A174503.

Programs

  • Magma
    I:=[2,10]; [n le 2 select I[n] else 10*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 07 2018
  • Mathematica
    a[0] = 2; a[1] = 10; a[n_] := 10a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 17}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{10,-1}, {2,10}, 30] (* G. C. Greubel, Nov 07 2018 *)
  • PARI
    polsym(x^2 - 10*x + 1,20) \\ Charles R Greathouse IV, Jun 11 2011
  • PARI
    {a(n) = 2 * real( (5 + 2 * quadgen(24))^n )}; /* Michael Somos, Feb 25 2014 */
  • Sage
    [lucas_number2(n,10,1) for n in range(27)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = (5+sqrt(24))^n + (5-sqrt(24))^n.
G.f.: (2-10*x)/(1-10*x+x^2). - Philippe Deléham, Nov 02 2008
From Peter Bala, Jan 06 2013: (Start)
Let F(x) = Product_{n = 0..inf} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 5 - sqrt(24). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.09989 80642 72052 68138 ... = 2 + 1/(10 + 1/(98 + 1/(970 + ...))).
Also F(-alpha) = 0.89989 78538 78393 34715 ... has the continued fraction representation 1 - 1/(10 - 1/(98 - 1/(970 - ...))) and the simple continued fraction expansion 1/(1 + 1/((10-2) + 1/(1 + 1/((98-2) + 1/(1 + 1/((970-2) + 1/(1 + ...))))))).
F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((10^2-4) + 1/(1 + 1/((98^2-4) + 1/(1 + 1/((970^2-4) + 1/(1 + ...))))))). Cf. A174503 and A005248. (End)
a(-n) = a(n). - Michael Somos, Feb 25 2014
From Peter Bala, Oct 16 2019: (Start)
8*Sum_{n >= 1} 1/(a(n) - 12/a(n)) = 1.
12*Sum_{n >= 1} (-1)^(n+1)/(a(n) + 8/a(n)) = 1.
Series acceleration formulas for sums of reciprocals:
Sum_{n >= 1} 1/a(n) = 1/8 - 12*Sum_{n >= 1} 1/(a(n)*(a(n)^2 - 12)) and
Sum_{n >= 1} (-1)^(n+1)/a(n) = 1/12 + 8*Sum_{n >= 1} (-1)^(n+1)/(a(n)*(a(n)^2 + 8)).
Sum_{n >= 1} 1/a(n) = ( (theta_3(5-sqrt(24)))^2 - 1 )/4 and
Sum_{n >= 1} (-1)^(n+1)/a(n) = ( 1 - (theta_3(sqrt(24)-5))^2 )/4, where theta_3(x) = 1 + 2*Sum_{n >= 1} x^(n^2) (see A000122). Cf. A153415 and A003499. (End)
E.g.f.: 2*exp(5*x)*cosh(2*sqrt(6)*x). - Stefano Spezia, Oct 18 2019
From Peter Bala, Mar 29 2022: (Start)
a(n) = 2*T(n,5), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind.
a(2^n) = A135927(n+1) and a(3^n) = A006242(n+1), both for n >= 0. (End)

Extensions

More terms from Colin Barker, Feb 25 2014

A036335 Total number of composite numbers with n digits and n prime factors (counted with multiplicity).

Original entry on oeis.org

0, 31, 225, 1563, 10222, 63030, 374264, 2160300, 12196405, 67724342, 371233523, 2014305995
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1998

Keywords

Comments

Essentially the same as A124033.

Examples

			a(1) = 0, since any single-digit number with 1 prime factor is a prime!

Links

Crossrefs

Cf. A036336, A036337, A036338.

Programs

  • Mathematica
    Table[Total[Table[If[CompositeQ[n]&&PrimeOmega[n]==x,1,0],{n,10^(x-1),10^x-1}]],{x,8}] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Jun 19 2022 *)

Extensions

One more term from Naohiro Nomoto, Jul 31 2001
a(9)-a(12) from Ray Chandler, Apr 12 2011

A036337 Largest integer with n digits and exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

7, 95, 994, 9999, 99996, 999992, 9999968, 99999840, 999999968, 9999999900, 99999999840, 999999999744, 9999999998720, 99999999998400, 999999999999000, 9999999999999744, 99999999999995904, 999999999999967232, 9999999999999989760, 99999999999999995904
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1998

Keywords

Comments

If all prime factors are distinct then a(n) >= A002110(n) which might give a contradiction for large enough n and so some primes have a multiplicity > k for some nonnegative k. - David A. Corneth, Oct 30 2018

Examples

			95 = 5 * 19, while 96, 97, 98, 99 and 100 have, respectively, 6,1,3,3 and 4 prime factors; thus 95 is the largest two digit number with exactly two prime factors.

Links

Crossrefs

Cf. A002110, A036335, A036336, A036338.

Programs

  • Mathematica
    Table[Module[{k=10^n-1},While[PrimeOmega[k]!=n,k--];k],{n,20}] (* Harvey P. Dale, Sep 02 2022 *)
  • PARI
    a(n) = forstep(i = 10^n-1,10^(n-1),-1,if(bigomega(i) == n, return(i))) \\ David A. Corneth, Oct 30 2018

Extensions

More terms and better description from Matthew Conroy, May 25 2001
a(19) and a(20) from Zak Seidov, Oct 30 2018

A036338 Composites whose digit length is equal to their number of prime factors (counted with multiplicity).

Original entry on oeis.org

10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1998

Keywords

Links

  • Carlos Rivera, Composed primes, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A036335, A036336, A036337.

Extensions

1 removed by Sean A. Irvine, Oct 26 2020

A342108 Smallest positive integer m with n digits and such that omega(m) = bigomega(m) = n.

Original entry on oeis.org

2, 10, 102, 1110, 10010, 101010, 1009470, 11741730, 223092870, 6469693230
Offset: 1

Views

Author

Bernard Schott, Feb 28 2021

Keywords

Comments

Equivalently: smallest n-digit squarefree number with n distinct prime factors.
Differs from A036336 where length(m) = bigomega(m) = n, when length(m) is the number of digits of m (A055642) and the n prime factors of m are counted with multiplicity (A001222).
Differs from A070842 where length(m) = omega(m) = n, when length(m) is the number of digits of m (A055642) and omega(m) is the number of distinct prime factors dividing m (A001221).
The first index for which these three sequences give three distinct terms is 4:
-> a(4) = 1110 = 2 * 3 * 5 * 37 , with length(1110) = omega(1110) = bigomega(1110) = 4.
-> A036336(4) = 1012 = 2 * 2 * 11 * 23 with length(1012) = bigomega(1012) = 4 > omega(1012) = 3.
-> A070842(4) = 1020 = 2 * 2 * 3 * 5 * 17 with length(1020) = omega(1020) = 4 < bigomega(1020) = 5.
As these terms are the smallest n-digit numbers in A167050 that is finite, this sequence is also finite with 10 terms, as for A070842.

Examples

			10010 = 2*5*7*11*13 is the smallest 5-digit number such that omega(10010) = bigomega(10010) = 5, hence a(5) = 10010.

Crossrefs

Subsequence of A167050.
Cf. A001221, A001222, A055642.
Cf. A036336, A070842, A342109.

Programs

  • Mathematica
    a={};For[n=1,n<=10,n++,For[m=10^(n-1),m<10^n,m++,If[PrimeOmega[m]==PrimeNu[m]==n,AppendTo[a,m];Break[]]]];a (* Stefano Spezia, Mar 04 2021 *)

Formula

A036336(n) <= A070842(n) <= a(n).
Showing 1-6 of 6 results.

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