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.

Previous Showing 31-37 of 37 results.

A332173 a(n) = 7*(10^(2n+1)-1)/9 - 4*10^n.

Original entry on oeis.org

3, 737, 77377, 7773777, 777737777, 77777377777, 7777773777777, 777777737777777, 77777777377777777, 7777777773777777777, 777777777737777777777, 77777777777377777777777, 7777777777773777777777777, 777777777777737777777777777, 77777777777777377777777777777, 7777777777777773777777777777777
Offset: 0

Views

Author

M. F. Hasler, Feb 06 2020

Keywords

Comments

According to M. Kamada, n = 0 and n = 2 are the only indices of a prime up to n = 2*10^4.

Links

Crossrefs

Cf. A138148 (cyclops numbers with binary digits only).
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

Programs

  • Maple
    A332173 := n -> 7*(10^(n*2+1)-1)/9 - 4*10^n;
  • Mathematica
    Array[7 (10^(2 # + 1) - 1)/9 - 4*10^# &, 15, 0]
  • PARI
    apply( {A332173(n)=10^(n*2+1)\9*7-4*10^n}, [0..15])
  • Python
    def A332173(n): return 10**(n*2+1)//9*7-4*10^n

Formula

a(n) = 7*A138148(n) + 3*10^n.
G.f.: (1 + 404*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.
E.g.f.: exp(x)*(70*exp(99*x) - 36*exp(9*x) - 7)/9. - Stefano Spezia, Feb 19 2025

A332174 a(n) = 7*(10^(2n+1)-1)/9 - 3*10^n.

Original entry on oeis.org

4, 747, 77477, 7774777, 777747777, 77777477777, 7777774777777, 777777747777777, 77777777477777777, 7777777774777777777, 777777777747777777777, 77777777777477777777777, 7777777777774777777777777, 777777777777747777777777777, 77777777777777477777777777777, 7777777777777774777777777777777
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

See A183179 = {2, 3, 6, 23, 36, 69, 561, ...} for the indices of primes.

Links

Crossrefs

Cf. A138148 (cyclops numbers with binary digits only).
Cf. (A077781-1)/2 = A183179: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

Programs

  • Maple
    A332174 := n -> 7*(10^(n*2+1)-1)/9 - 3*10^n;
  • Mathematica
    Array[7 (10^(2 # + 1) - 1)/9 - 3*10^# &, 15, 0]
  • PARI
    apply( {A332174(n)=10^(n*2+1)\9*7-3*10^n}, [0..15])
  • Python
    def A332174(n): return 10**(n*2+1)//9*7-3*10^n

Formula

a(n) = 7*A138148(n) + 4*10^n.
G.f.: (4 + 303*x - 1000*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.: (1/9)*exp(x)*(70*exp(99*x) - 27*exp(9*x) - 7). - Stefano Spezia, Feb 08 2020

A332175 a(n) = 7*(10^(2n+1)-1)/9 - 2*10^n.

Original entry on oeis.org

5, 757, 77577, 7775777, 777757777, 77777577777, 7777775777777, 777777757777777, 77777777577777777, 7777777775777777777, 777777777757777777777, 77777777777577777777777, 7777777777775777777777777, 777777777777757777777777777, 77777777777777577777777777777, 7777777777777775777777777777777
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

See A183180 = {0, 1, 7, 13, 58, 129, 253, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077785-1)/2 = A183180: indices of primes.
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

Programs

  • Maple
    A332175 := n -> 7*(10^(n*2+1)-1)/9 - 2*10^n;
  • Mathematica
    Array[7 (10^(2 # + 1) - 1)/9 - 2*10^# &, 15, 0]
  • PARI
    apply( {A332175(n)=10^(n*2+1)\9*7-2*10^n}, [0..15])
  • Python
    def A332175(n): return 10**(n*2+1)//9*7-2*10^n

Formula

a(n) = 7*A138148(n) + 5*10^n.
G.f.: (5 + 202*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.
E.g.f.: (1/9)*exp(x)*(70*exp(99*x) - 18*exp(9*x) - 7). - Stefano Spezia, Feb 08 2020

A332176 a(n) = 7*(10^(2n+1)-1)/9 - 10^n.

Original entry on oeis.org

6, 767, 77677, 7776777, 777767777, 77777677777, 7777776777777, 777777767777777, 77777777677777777, 7777777776777777777, 777777777767777777777, 77777777777677777777777, 7777777777776777777777777, 777777777777767777777777777, 77777777777777677777777777777, 7777777777777776777777777777777
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

See A183181 = {4, 5, 8, 11, 1244, 1685, ...} for the indices of primes.

Links

Crossrefs

Cf. A138148 (cyclops numbers with binary digits only).
Cf. (A077788-1)/2 = A183181: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

Programs

  • Maple
    A332176 := n -> 7*(10^(n*2+1)-1)/9 - 10^n;
  • Mathematica
    Array[7 (10^(2 # + 1) - 1)/9 - 10^# &, 15, 0]
  • PARI
    apply( {A332176(n)=10^(n*2+1)\9*7-10^n}, [0..15])
  • Python
    def A332176(n): return 10**(n*2+1)//9*7-10^n

Formula

a(n) = 7*A138148(n) + 6*10^n.
G.f.: (6 + 101*x - 800*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.

A363377 Largest positive integer having n holes that can be made using the fewest possible digits.

Original entry on oeis.org

7, 9, 8, 98, 88, 988, 888, 9888, 8888, 98888, 88888, 988888, 888888, 9888888, 8888888, 98888888, 88888888, 988888888, 888888888, 9888888888, 8888888888, 98888888888, 88888888888, 988888888888, 888888888888, 9888888888888, 8888888888888, 98888888888888, 88888888888888, 988888888888888
Offset: 0

Views

Author

Julia Zimmerman, May 29 2023

Keywords

Comments

Each decimal digit has 0, 1 or 2 holes so that n holes requires A065033(n) digits.

Examples

			For n=0, the largest integer with no holes in it that is as short as possible is 7 (9 is larger, but has 1 hole; 11 is larger and has no holes, but is longer at length 2 > length 1).
For n=1, the largest integer with 1 hole that is as short as possible is 9 (following the same kind of reasoning as with n=0).

Links

Crossrefs

Cf. A002281 and A002282 (number of holes), A065033 (digits required).
Cf. A249572 and A250256 (smallest number).
Cf. A337099 (largest 7-segment).

Programs

  • Mathematica
    CoefficientList[Series[(7 + 2 x - 71 x^2 + 70 x^3)/((1 - x) (1 - 10 x^2)), {x, 0, 30}], x] (* Michael De Vlieger, Jul 05 2023 *)
  • Python
    A363377=lambda n: (8+n%2*81)*10**(n>>1)//9 if n else 7
print([A363377(n) for n in range(30)]) # Natalia L. Skirrow, Jun 26 2023

Formula

From Natalia L. Skirrow, Jun 26 2023: (Start)
a(n) = (89*(10^((n-1)/2))-8)/9 for odd n; a(n) = 8*(10^(n/2)-1)/9 for even n >= 2.
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3), for n >= 4.
G.f.: (7+2*x-71*x^2+70*x^3)/((1-x)*(1-10*x^2)).
E.g.f.: (80*cosh(sqrt(10)*x) + 89*sqrt(10)*sinh(sqrt(10)*x) - 80*e^x)/90 + 7. (End)

A205087 a(n) = n 7's sandwiched between two 1's.

Original entry on oeis.org

11, 171, 1771, 17771, 177771, 1777771, 17777771, 177777771, 1777777771, 17777777771, 177777777771, 1777777777771, 17777777777771, 177777777777771, 1777777777777771, 17777777777777771, 177777777777777771, 1777777777777777771, 17777777777777777771
Offset: 0

Views

Author

José María Grau Ribas, Jan 22 2012

Keywords

Links

Crossrefs

Cf. A002281.

Programs

  • Mathematica
    a[0]=11;a[n_]:=a[n-1]*10+61;Table[a[n],{n,0,44}]
Table[10 FromDigits[PadRight[{1},n,7]]+1,{n,20}] (* Harvey P. Dale, Nov 18 2022 *)
  • PARI
    a(n)=(160*10^n-61)/9 \\ Charles R Greathouse IV, Jan 23 2012

Formula

a(0)=1, a(n) = 10*a(n-1) + 61.
a(n) = (160*10^n - 61)/9 (see PARI's code by Charles R Greathouse IV).
From Elmo R. Oliveira, Feb 18 2025: (Start)
G.f.: (11 + 50*x)/((1 - x)*(1 - 10*x)).
E.g.f.: exp(x)*(160*exp(9*x) - 61)/9.
a(n) = 11*a(n-1) - 10*a(n-2). (End)

A361820 Palindromes in A329150.

Original entry on oeis.org

0, 2, 3, 5, 7, 11, 22, 33, 55, 77, 202, 222, 232, 252, 272, 303, 313, 323, 333, 353, 373, 505, 525, 535, 555, 575, 707, 717, 727, 737, 757, 777, 1111, 2002, 2112, 2222, 2332, 2552, 2772, 3003, 3113, 3223, 3333, 3553, 3773, 5005, 5115, 5225, 5335, 5555, 5775, 7007, 7117
Offset: 1

Views

Author

Bernard Schott, Mar 25 2023

Keywords

Comments

If m is a palindrome with no digit greater than 5 in A118597, then A329147(m) is a term, but there exist terms that are not of this form as 313, 717, ...

Examples

			232 is a term which has two preimages since A329147(91) = A329147(121) = 232.
313 = A329147(26) is a term whose preimage is not in A118597.
2002 is a term since A329147(1001) = 2002.
2112 is a term since A329147(151) = 2112.
27172 = A329147(1471) is a term whose preimage is not in A118597.

Crossrefs

Intersection of A002113 and A329150.
Cf. A118597, A329147, A361750, A361821.
Subsequences: A002276, A002277, A002279, A002281, A099814.

Programs

  • Mathematica
    p[n_] := If[n > 0, Prime[n], 0]; seq[ndigmax_] := Module[{t = Table[FromDigits[ Flatten@ IntegerDigits@ (p /@ IntegerDigits[n])], {n, 0, 10^ndigmax - 1}]}, Union@ Select[t, # < 10^ndigmax && PalindromeQ[#] &]]; seq[4] (* Amiram Eldar, Mar 26 2023 *)
  • PARI
    ispal(n) = my(d=digits(n)); d==Vecrev(d);
f(n) = if (n, fromdigits(concat(apply(d -> if (d, digits(prime(d)), [0]), digits(n)))), 0); \\ A329147
lista(nn) = my(list = List(), m); for (n=0, nn, m = f(n); if ((m <= nn) && ispal(m), listput(list, m));); vecsort(Set(list)); \\ Michel Marcus, Mar 26 2023
Previous Showing 31-37 of 37 results.

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