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 1-10 of 10 results.

A332112 a(n) = (10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

2, 121, 11211, 1112111, 111121111, 11111211111, 1111112111111, 111111121111111, 11111111211111111, 1111111112111111111, 111111111121111111111, 11111111111211111111111, 1111111111112111111111111, 111111111111121111111111111, 11111111111111211111111111111, 1111111111111112111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

a(0) = 2 is the only prime in this sequence, since all other terms factor as a(n) = R(n+1)*(10^n+1), where R(n) = (10^n-1)/9.

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332132 .. A332192 (variants with different repeated digit 3, ..., 9).
Cf. A332113 .. A332119 (variants with different middle digit 3, ..., 9).
Cf. A331860 & A331861 (indices of primes in non-palindromic variants).

Programs

  • Maple
    A332112 := n -> (10^(2*n+1)-1)/9+10^n;
  • Mathematica
    Array[ (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
  • PARI
    apply( {A332112(n)=10^(n*2+1)\9*1+10^n}, [0..15])
  • Python
    def A332112(n): return 10**(n*2+1)//9+10**n

Formula

a(n) = A138148(n) + 2*10^n = A002275(2n+1) + 10^n.
G.f.: (2 - 101*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332139 a(n) = (10^(2*n+1)-1)/3 + 6*10^n.

Original entry on oeis.org

9, 393, 33933, 3339333, 333393333, 33333933333, 3333339333333, 333333393333333, 33333333933333333, 3333333339333333333, 333333333393333333333, 33333333333933333333333, 3333333333339333333333333, 333333333333393333333333333, 33333333333333933333333333333, 3333333333333339333333333333333
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332129 .. A332189 (variants with different repeated digit 2, ..., 8).
Cf. A332130 .. A332138 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332139 := n -> (10^(2*n+1)-1)/3+6*10^n;
  • Mathematica
    Array[ (10^(2 # + 1)-1)/3 + 6*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{9,393,33933},20] (* Harvey P. Dale, Sep 17 2020 *)
  • PARI
    apply( {A332139(n)=10^(n*2+1)\3+6*10^n}, [0..15])
  • Python
    def A332139(n): return 10**(n*2+1)//3+6*10**n

Formula

a(n) = 3*A138148(n) + 9*10^n = A002277(2n+1) + 6*10^n = 3*A332113(n).
G.f.: (9 - 606*x + 300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332193 a(n) = 10^(2n+1) - 1 - 6*10^n.

Original entry on oeis.org

3, 939, 99399, 9993999, 999939999, 99999399999, 9999993999999, 999999939999999, 99999999399999999, 9999999993999999999, 999999999939999999999, 99999999999399999999999, 9999999999993999999999999, 999999999999939999999999999, 99999999999999399999999999999, 9999999999999993999999999999999
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332113 .. A332183 (variants with different repeated digit 1, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332193 := n -> 10^(n*2+1)-1-6*10^n;
  • Mathematica
    Array[ 10^(2 # + 1) - 1 - 6*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{3,939,99399},20] (* Harvey P. Dale, Jan 19 2024 *)
  • PARI
    apply( {A332193(n)=10^(n*2+1)-1-6*10^n}, [0..15])
  • Python
    def A332193(n): return 10**(n*2+1)-1-6*10^n

Formula

a(n) = 9*A138148(n) + 3*10^n = A002283(2n+1) - 6*10^n.
G.f.: (3 + 606*x - 1500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332123 a(n) = 2*(10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

3, 232, 22322, 2223222, 222232222, 22222322222, 2222223222222, 222222232222222, 22222222322222222, 2222222223222222222, 222222222232222222222, 22222222222322222222222, 2222222222223222222222222, 222222222222232222222222222, 22222222222222322222222222222, 2222222222222223222222222222222
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332123 := n -> 2*(10^(2*n+1)-1)/9+10^n;
  • Mathematica
    Array[2 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
  • PARI
    apply( {A332123(n)=10^(n*2+1)\9*2+10^n}, [0..15])
  • Python
    def A332123(n): return 10**(n*2+1)//9*2+10**n

Formula

a(n) = 2*A138148(n) + 3*10^n = A002276(2n+1) + 10^n.
G.f.: (3 - 101*x - 100*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A077779 Numbers k such that (10^k - 1)/9 + 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 5, 39, 195, 19637
Offset: 1

Views

Author

Patrick De Geest, Nov 16 2002

Keywords

Comments

Prime versus probable prime status and proofs are given in the author's table.
a(6) > 2*10^5. - Robert Price, Apr 02 2016
The number k = 1 would also correspond to a prime, 3, but not "near-repdigit" or "wing" in a strict sense. - M. F. Hasler, Feb 09 2020

Examples

			5 is a term because (10^5 - 1)/9 + 2*10^2 = 11311.

References

  • C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.

Links

Crossrefs

Cf. A004023, A077775-A077798, A107123-A107127, A107648, A107649, A115073, A183174-A183187.
See A332113 for the (prime and composite) near-repunit palindromes 1..131..1.

Programs

  • Mathematica
    Do[ If[ PrimeQ[(10^n + 18*10^Floor[n/2] - 1)/9], Print[n]], {n, 3, 20000, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

Formula

a(n) = 2*A107123(n+1) + 1.

Extensions

Name corrected by Jon E. Schoenfield, Oct 31 2018

A332126 a(n) = 2*(10^(2n+1)-1)/9 + 4*10^n.

Original entry on oeis.org

6, 262, 22622, 2226222, 222262222, 22222622222, 2222226222222, 222222262222222, 22222222622222222, 2222222226222222222, 222222222262222222222, 22222222222622222222222, 2222222222226222222222222, 222222222222262222222222222, 22222222222222622222222222222, 2222222222222226222222222222222
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332116 .. A332196 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332126 := n -> 2*(10^(2*n+1)-1)/9+4*10^n;
  • Mathematica
    Array[2 (10^(2 # + 1)-1)/9 + 4*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,2],{6},PadRight[{},n,2]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{6,262,22622},20] (* Harvey P. Dale, Oct 17 2021 *)
  • PARI
    apply( {A332126(n)=10^(n*2+1)\9*2+4*10^n}, [0..15])
  • Python
    def A332126(n): return 10**(n*2+1)//9*2+4*10**n

Formula

a(n) = 2*A138148(n) + 6*10^n = A002276(2n+1) + 4*10^n = 2*A332113(n).
G.f.: (6 - 404*x + 200*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: 2*exp(x)*(10*exp(99*x) + 18*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A332143 a(n) = 4*(10^(2*n+1)-1)/9 - 10^n.

Original entry on oeis.org

3, 434, 44344, 4443444, 444434444, 44444344444, 4444443444444, 444444434444444, 44444444344444444, 4444444443444444444, 444444444434444444444, 44444444444344444444444, 4444444444443444444444444, 444444444444434444444444444, 44444444444444344444444444444, 4444444444444443444444444444444
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002278 (4*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332143 := n -> 4*(10^(2*n+1)-1)/9-10^n;
  • Mathematica
    Array[4 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
  • PARI
    apply( {A332143(n)=10^(n*2+1)\9*4-10^n}, [0..15])
  • Python
    def A332143(n): return 10**(n*2+1)//9*4-10**n

Formula

a(n) = 4*A138148(n) + 3*10^n = A002278(2n+1) - 10^n.
G.f.: (3 + 101*x - 500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332183 a(n) = 8*(10^(2n+1)-1)/9 - 5*10^n.

Original entry on oeis.org

3, 838, 88388, 8883888, 888838888, 88888388888, 8888883888888, 888888838888888, 88888888388888888, 8888888883888888888, 888888888838888888888, 88888888888388888888888, 8888888888883888888888888, 888888888888838888888888888, 88888888888888388888888888888, 8888888888888883888888888888888
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332183 := n -> 8*(10^(2*n+1)-1)/9-5*10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 5*10^# &, 15, 0]
  • PARI
    apply( {A332183(n)=10^(n*2+1)\9*8-5*10^n}, [0..15])
  • Python
    def A332183(n): return 10**(n*2+1)//9*8-5*10**n

Formula

a(n) = 8*A138148(n) + 3*10^n = A002282(2n+1) - 5*10^n.
G.f.: (3 + 505*x - 1300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332153 a(n) = 5*(10^(2*n+1)-1)/9 - 2*10^n.

Original entry on oeis.org

3, 535, 55355, 5553555, 555535555, 55555355555, 5555553555555, 555555535555555, 55555555355555555, 5555555553555555555, 555555555535555555555, 55555555555355555555555, 5555555555553555555555555, 555555555555535555555555555, 55555555555555355555555555555, 5555555555555553555555555555555
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332153 := n -> 5*(10^(2*n+1)-1)/9-2*10^n;
  • Mathematica
    Array[5 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]
  • PARI
    apply( {A332153(n)=10^(n*2+1)\9*5-2*10^n}, [0..15])
  • Python
    def A332153(n): return 10**(n*2+1)//9*5-2*10**n

Formula

a(n) = 5*A138148(n) + 3*10^n = A002279(2n+1) - 2*10^n.
G.f.: (3 + 202*x - 700*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332163 a(n) = 6*(10^(2*n+1)-1)/9 - 3*10^n.

Original entry on oeis.org

3, 636, 66366, 6663666, 666636666, 66666366666, 6666663666666, 666666636666666, 66666666366666666, 6666666663666666666, 666666666636666666666, 66666666666366666666666, 6666666666663666666666666, 666666666666636666666666666, 66666666666666366666666666666, 6666666666666663666666666666666
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002280 (6*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9).
Cf. A332160 .. A332169 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332163 := n -> 6*(10^(2*n+1)-1)/9-3*10^n;
  • Mathematica
    Array[6 (10^(2 # + 1)-1)/9 - 3*10^# &, 15, 0]
  • PARI
    apply( {A332163(n)=10^(n*2+1)\9*6-3*10^n}, [0..15])
  • Python
    def A332163(n): return 10**(n*2+1)//9*6-3*10**n

Formula

a(n) = 6*A138148(n) + 3*10^n = A002280(2n+1) - 3*10^n = 3*A332121(n).
G.f.: (3 + 303*x - 900*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
Showing 1-10 of 10 results.

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