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.

User: Ravi Bhandari

Ravi Bhandari's wiki page.

Ravi Bhandari has authored 5 sequences.

A341828 Difference of consecutive odd squarefree semiprimes.

Original entry on oeis.org

6, 12, 2, 4, 12, 4, 2, 8, 4, 8, 8, 2, 4, 2, 2, 16, 4, 4, 4, 6, 4, 8, 2, 2, 10, 4, 2, 16, 6, 2, 2, 14, 2, 2, 4, 4, 2, 2, 2, 2, 14, 2, 10, 2, 4, 6, 6, 2, 20, 4, 4, 4, 2, 2, 2, 4, 10, 2, 2, 4, 2, 6, 4, 2, 14, 10, 6, 6, 4, 10, 2, 2, 8, 4, 4, 2, 2, 2, 10, 10, 8, 2, 4
Offset: 1

Views

Author

Ravi Bhandari, Feb 20 2021

Keywords

Crossrefs

Cf. A006881, A046388.

Programs

  • PARI
    lista(nn) ={my(r=15); forstep(k=17, nn, 2, if(bigomega(k)==2 && issquarefree(k), print1(k-r, ", "); r=k)); } \\ Jinyuan Wang, Mar 01 2021

Extensions

More terms from Jinyuan Wang, Mar 01 2021

A216754 Digital root of fourth power of Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9
Offset: 0

Views

Author

Ravi Bhandari, Sep 15 2012

Keywords

Comments

This sequence is periodic with period 12. Also, the first (2n - 1) terms are symmetric about n-th term, where n = 6k, k = 1, 2, 3, ...

Links

Crossrefs

Cf. A004090, A216676, A216699.

Programs

  • Mathematica
    Table[NestWhile[Total[IntegerDigits[#]] &, Fibonacci[n]^4, # > 9 &], {n, 0, 86}] (* T. D. Noe, Oct 15 2012 *)
Join[{0},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9},108]] (* Ray Chandler, Aug 27 2015 *)

Extensions

Extended by Ray Chandler, Aug 27 2015

A216699 Digital root of cubes of Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8
Offset: 0

Views

Author

Ravi Bhandari, Sep 15 2012

Keywords

Comments

This sequence repeats after every 8 terms, hence this is periodic with period 8.

Links

Crossrefs

Cf. A004090, A056570, A216676.

Programs

  • Mathematica
    Table[NestWhile[Total[IntegerDigits[#]] &, Fibonacci[n]^3, # > 9 &], {n, 0, 86}] (* T. D. Noe, Oct 15 2012 *)
Join[{0},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{1, 1, 8, 9, 8, 8, 1, 9},86]] (* Ray Chandler, Aug 25 2015 *)

A216755 Digital root of the fifth power of Fibonacci(n).

Original entry on oeis.org

1, 1, 5, 9, 2, 8, 7, 9, 4, 1, 8, 9, 8, 8, 4, 9, 7, 1, 2, 9, 5, 8, 1, 9, 1, 1, 5, 9, 2, 8, 7, 9, 4, 1, 8, 9, 8, 8, 4, 9, 7, 1, 2, 9, 5, 8, 1, 9, 1, 1, 5, 9, 2, 8, 7, 9, 4, 1, 8, 9, 8, 8, 4, 9, 7, 1, 2, 9, 5, 8, 1, 9, 1, 1, 5, 9, 2, 8, 7, 9, 4, 1, 8, 9, 8, 8, 4, 9, 7, 1, 2, 9, 5, 8, 1, 9, 1, 1, 5, 9
Offset: 1

Views

Author

Ravi Bhandari, Sep 15 2012

Keywords

Comments

This sequence is periodic with period 24, i.e. gcd(period of digital roots of squares of Fibonacci, period of digital roots of cubes of Fibonacci)

Links

Crossrefs

Cf. A030132, A216676, A216699

Programs

  • Mathematica
    (* First run program for A211821 to define digitalRoot *) Table[digitalRoot[Fibonacci[n]^5], {n, 90}] (* Alonso del Arte, Sep 15 2012 *)
LinearRecurrence[{0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1},{1, 1, 5, 9, 2, 8, 7, 9, 4, 1, 8, 9, 8, 8, 4, 9},100] (* Ray Chandler, Aug 27 2015 *)

Formula

a(n) = A010888(A056572(n)).
a(n) = a(n-4) - a(n-12) + a(n-16). - R. J. Mathar, Sep 15 2012
G.f. x*( -1-x-5*x^2-9*x^3-x^4-7*x^5-2*x^6-2*x^8+7*x^9-x^10-5*x^12-8*x^13-x^14-9*x^15 ) / ( (x-1) *(1+x) *(x^2+1) *(x^4+1) *(x^8-x^4+1) ). - R. J. Mathar, Sep 15 2012

A216676 Digital roots of squares of Fibonacci numbers.

Original entry on oeis.org

1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4
Offset: 1

Views

Author

Ravi Bhandari, Sep 14 2012

Keywords

Comments

The first 11 terms are symmetric about 6th term. The first 23 terms are symmetric about 12th term. We can generalize this as follows: the first (2n-1) terms are symmetric about n-th term.
The sequence appears to be periodic with period-length 12. - John W. Layman, Sep 14 2012
The Fibonacci numbers are periodic modulo any integer. The digital roots of the Fibonacci numbers are given by A030132, a sequence with a period length of 24. Squaring gives {1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9}, which is a sequence of twelve numbers given twice. Therefore, the previous comment is correct. - Alonso del Arte, Sep 25 2012

Examples

			a(7) = 7 because F(7) = 13 and 13^2 = 169, with digits adding up to 16, the digital root is therefore 7.

Links

Programs

  • Mathematica
    a = {}; For[n = 1, n <= 100, n++, {fn2 = Fibonacci[n]^2; d = IntegerDigits[fn2]; While[Length[d] > 1, d = IntegerDigits[Total[d]]]; AppendTo[a, d[[1]]] }]; a (* John W. Layman,  Sep 14 2012 *)
ReplaceAll[Table[Mod[Fibonacci[n]^2, 9], {n, 72}], {0 -> 9}] (* Alonso del Arte, Sep 23 2012 *)
  • PARI
    fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
a(n)=lift(fibmod(n,9)^2-1)+1 \\ Charles R Greathouse IV, Jun 20 2017

Formula

a(n) = A010888(A007598(n)).
G.f. ( -1-x-3*x^2-8*x^3-3*x^4+8*x^5-9*x^7-x^6 ) / ( (x-1) *(1+x) *(x^2+1) *(x^4-x^2+1) ). - R. J. Mathar, Sep 15 2012

Extensions

Terms a(25)-a(72) by John W. Layman, Sep 14 2012
Terms a(73) and beyond from Andrew Howroyd, Feb 25 2018

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