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.

A332197 a(n) = 10^(2n+1) - 1 - 2*10^n.

Original entry on oeis.org

7, 979, 99799, 9997999, 999979999, 99999799999, 9999997999999, 999999979999999, 99999999799999999, 9999999997999999999, 999999999979999999999, 99999999999799999999999, 9999999999997999999999999, 999999999999979999999999999, 99999999999999799999999999999, 9999999999999997999999999999999
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

According to Kamada, n = 118 and n = 145126 are the only known indices of primes (the so-called palindromic near-repdigit or wing primes).

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).
Cf. A332190 .. A332196, A181965 (variants with different middle digit 0, ..., 8).
Cf. A332117 .. A332187 (variants with different repeated digit 1, ..., 9).

Programs

  • Maple
    A332197 := n -> 10^(n*2+1)-1-2*10^n;
  • Mathematica
    Array[ 10^(2 # + 1) -1 -2*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,9],{7},PadRight[{},n,9]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{7,979,99799},20] (* Harvey P. Dale, Mar 03 2023 *)
  • PARI
    apply( {A332197(n)=10^(n*2+1)-1-2*10^n}, [0..15])
  • Python
    def A332197(n): return 10**(n*2+1)-1-2*10^n

Formula

a(n) = 9*A138148(n) + 7*10^n.
G.f.: (7 + 202*x - 1100*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.

A332127 a(n) = 2*(10^(2n+1)-1)/9 + 5*10^n.

Original entry on oeis.org

7, 272, 22722, 2227222, 222272222, 22222722222, 2222227222222, 222222272222222, 22222222722222222, 2222222227222222222, 222222222272222222222, 22222222222722222222222, 2222222222227222222222222, 222222222222272222222222222, 22222222222222722222222222222, 2222222222222227222222222222222
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. A332117 .. A332197 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332127 := n -> 2*(10^(2*n+1)-1)/9+5*10^n;
  • Mathematica
    Array[2 (10^(2 # + 1)-1)/9 + 5*10^# &, 15, 0]
  • PARI
    apply( {A332127(n)=10^(n*2+1)\9*2+5*10^n}, [0..15])
  • Python
    def A332127(n): return 10**(n*2+1)//9*2+5*10**n

Formula

a(n) = 2*A138148(n) + 7*10^n = A002276(2n+1) + 5*10^n.
G.f.: (7 - 505*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.

A332187 a(n) = 8*(10^(2n+1)-1)/9 - 10^n.

Original entry on oeis.org

7, 878, 88788, 8887888, 888878888, 88888788888, 8888887888888, 888888878888888, 88888888788888888, 8888888887888888888, 888888888878888888888, 88888888888788888888888, 8888888888887888888888888, 888888888888878888888888888, 88888888888888788888888888888, 8888888888888887888888888888888
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Links

Crossrefs

Cf. (A077776-1)/2 = A183190: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332117 .. A332197 (variants with different "wing" digit 1, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332187 := n -> 8*(10^(2*n+1)-1)/9-10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{7,878,88788},20] (* Harvey P. Dale, Jul 21 2024 *)
  • PARI
    apply( {A332187(n)=10^(n*2+1)\9*8-10^n}, [0..15])
  • Python
    def A332187(n): return 10**(n*2+1)//9*8-10**n

Formula

a(n) = 8*A138148(n) + 7*10^n = A002282(2n+1) - 10^n.
G.f.: (7 + 101*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.

A332167 a(n) = 6*(10^(2*n+1)-1)/9 + 10^n.

Original entry on oeis.org

7, 676, 66766, 6667666, 666676666, 66666766666, 6666667666666, 666666676666666, 66666666766666666, 6666666667666666666, 666666666676666666666, 66666666666766666666666, 6666666666667666666666666, 666666666666676666666666666, 66666666666666766666666666666, 6666666666666667666666666666666
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. A332117 .. A332197 (variants with different repeated digit 1, ..., 9).
Cf. A332160 .. A332169 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332167 := n -> 6*(10^(2*n+1)-1)/9+10^n;
  • Mathematica
    Array[6 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
  • PARI
    apply( {A332167(n)=10^(n*2+1)\9*6+10^n}, [0..15])
  • Python
    def A332167(n): return 10**(n*2+1)//9*6+10**n

Formula

a(n) = 6*A138148(n) + 7*10^n = A002280(2n+1) + 10^n.
G.f.: (7 - 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.

A332147 a(n) = 4*(10^(2*n+1)-1)/9 + 3*10^n.

Original entry on oeis.org

7, 474, 44744, 4447444, 444474444, 44444744444, 4444447444444, 444444474444444, 44444444744444444, 4444444447444444444, 444444444474444444444, 44444444444744444444444, 4444444444447444444444444, 444444444444474444444444444, 44444444444444744444444444444, 4444444444444447444444444444444
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. A332117 .. A332197 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332147 := n -> 4*(10^(2*n+1)-1)/9+3*10^n;
  • Mathematica
    Array[4 (10^(2 # + 1)-1)/9 + 3*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{7,474,44744},20] (* Harvey P. Dale, Mar 08 2022 *)
  • PARI
    apply( {A332147(n)=10^(n*2+1)\9*4+3*10^n}, [0..15])
  • Python
    def A332147(n): return 10**(n*2+1)//9*4+3*10**n

Formula

a(n) = 4*A138148(n) + 7*10^n = A002278(2n+1) + 3*10^n.
G.f.: (7 - 303*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.

A332157 a(n) = 5*(10^(2*n+1)-1)/9 + 2*10^n.

Original entry on oeis.org

7, 575, 55755, 5557555, 555575555, 55555755555, 5555557555555, 555555575555555, 55555555755555555, 5555555557555555555, 555555555575555555555, 55555555555755555555555, 5555555555557555555555555, 555555555555575555555555555, 55555555555555755555555555555, 5555555555555557555555555555555
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. A332117 .. A332197 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332157 := 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( {A332157(n)=10^(n*2+1)\9*5+2*10^n}, [0..15])
  • Python
    def A332157(n): return 10**(n*2+1)//9*5+2*10**n

Formula

a(n) = 5*A138148(n) + 7*10^n = A002279(2n+1) + 2*10^n.
G.f.: (7 - 202*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.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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