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 1-7 of 7 results.

A028416 Primes p such that the decimal expansion of 1/p has a periodic part of even length.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 29, 47, 59, 61, 73, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 193, 197, 211, 223, 229, 233, 241, 251, 257, 263, 269, 281, 293, 313, 331, 337, 349, 353, 367, 373, 379, 383, 389, 401, 409, 419, 421, 433
Offset: 1

Views

Author

Mario Velucchi (mathchess(AT)velucchi.it)

Keywords

Comments

Primes whose reciprocals have even period length.
Primes p such that the order of 10 mod p is even. - Joerg Arndt, Mar 04 2014
A002371(A049084(a(n))) mod 2 == 0.
Not the same as A040121: a(33)=241 is not in A040121.
Let (d(i): 1<=i<=2*K) be the period of the decimal expansion of 1/a(n), K=A002371(A049084(a(n)))/2, then d(i) + d(i+K) = 9 for i with 1<=i<=K, or, equivalently: u + v = 10^K - 1 with u = Sum_{i=1..K} d(i)*10^(K-i) and v = Sum_{i=1..K} d(i+K)*10^(K-i). - Reinhard Zumkeller, Oct 05 2008

Examples

			From _Reinhard Zumkeller_, Oct 05 2008: (Start)
(0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7) is the period of 1/17 (see A007450),
K = A002371(A049084(17))/2 = A002371(7)/2 = 16/2 = 8,
u = 5882352, v = 94117647: u + v = 99999999 = 10^8 - 1. (End)

References

  • H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, "Die periodischen Dezimalbrueche". [Reinhard Zumkeller, Oct 05 2008]

Links

Crossrefs

Cf. A087000, A186635.

Programs

  • Maple
    A028416 := proc(n) local st:
st := ithprime(n):
if (modp(numtheory[order](10,st),2) = 0) then
   RETURN(st)
fi: end:  seq(A028416(n), n=1..100); # Jani Melik, Feb 24 2011
  • Mathematica
    Select[Prime[Range[4,100]],EvenQ[Length[RealDigits[1/#][[1,1]]]]&] (* Harvey P. Dale, Jul 07 2011 *)
  • PARI
    forprime(p=7,1e3,if(znorder(Mod(10,p))%2==0,print1(p", "))) \\ Charles R Greathouse IV, Feb 24 2011
  • Python
    from sympy import gcd, isprime, n_order
is_A028416 = lambda n: gcd(n,10)==1 and n>5 and n_order(10, n)%2==0 and isprime(n) # M. F. Hasler, Nov 19 2024

Extensions

More terms from Reinhard Zumkeller, Jul 29 2003

A242828 Primes formed by the initial digits of the decimal expansion of 1/17, starting at the first nonzero digit in the expansion.

Original entry on oeis.org

5, 5882352941, 588235294117, 588235294117647058823529411764705882352941176470588235294117
Offset: 1

Views

Author

Felix Fröhlich, May 23 2014

Keywords

Comments

There is no other term with 126 or fewer digits.
No more terms < 10^20000. - Jon E. Schoenfield, Nov 02 2019

Crossrefs

Cf. A007450.
Corresponding sequences for 1/k: A242824 (k=7), A093676 (k=12), A242826 (k=13), A242827(k=14), A242833 (k=19).

Programs

  • Mathematica
    Select[Table[FromDigits[PadRight[{},n,{5,8,8,2,3,5,2,9,4,1,1,7,6,4,7,0}]],{n,60}],PrimeQ] (* Harvey P. Dale, Aug 08 2021 *)
  • PARI
    lista(nn) = {v = [5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0]; n = 0; for (i=0, nn, n = 10*n+ v[(i % 16)+1]; if (ispseudoprime(n), print1(#Str(n), ", ")););} \\ Michel Marcus, May 27 2014

A236184 Decimal expansion of 1/65537.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 2, 5, 8, 5, 5, 6, 2, 3, 5, 4, 0, 9, 0, 0, 5, 5, 9, 9, 8, 9, 0, 1, 3, 8, 3, 9, 5, 1, 0, 5, 0, 5, 5, 1, 5, 9, 6, 8, 0, 7, 9, 1, 0, 0, 3, 5, 5, 5, 2, 4, 3, 6, 0, 2, 8, 5, 0, 2, 9, 8, 3, 0, 4, 7, 7, 4, 4, 0, 2, 2, 4, 6, 0, 5, 9, 4, 7, 7, 8, 5, 2, 2, 0, 5, 6, 2, 4, 3, 0, 3, 8, 2, 8, 3, 7, 1, 7, 5, 9
Offset: 0

Views

Author

Rick L. Shepherd, Jan 19 2014

Keywords

Comments

Periodic sequence of period 65536. Being a Fermat prime > 5, 65537 is a full reptend prime. One full period is given in the table.

Examples

			0.000015258556235409005599890138395105055159680791003555243602850298304774...

Links

Crossrefs

Cf. A007450 (1/17), A021261 (1/257), A019434, A001913, A048963.

Programs

  • Mathematica
    RealDigits[1/65537, 10, 105, -1][[1]] (* T. D. Noe, Jan 27 2014 *)
  • PARI
    {default(realprecision, 66000);
x = 1/65537; d = 0; for(n = 0, 65535, x = (x-d)*10; d = floor(x);
write("b236184.txt", n, " ", d))} \\ Rick L. Shepherd, Jan 19 2014 (after similar program by Harry J. Smith)

Formula

a(n + 65536) = a(n).

A303816 Decimal expansion of 2700/17.

Original entry on oeis.org

1, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9
Offset: 3

Views

Author

Omar E. Pol, Jun 13 2018

Keywords

Comments

Decimal expansion of the internal angle of the regular heptadecagon (in degrees).
Period 16. - Jianing Song, Jun 22 2018

Examples

			158.82352941176470588235294117647058823529411764705882352941176470...

Links

Crossrefs

Essentially the same as A007450. Cf. A019434, A210644, A210649, A303817.

Formula

Equals 180*15/17 = 158 + (14/17) = 180 - A303817.
From Chai Wah Wu, Dec 20 2019: (Start)
a(n) = a(n-1) - a(n-8) + a(n-9) for n > 12.
G.f.: x^3*(x^9 - 8*x^8 + 3*x^7 - 2*x^6 - x^5 + 6*x^4 - 3*x^2 - 4*x - 1)/((x - 1)*(x^8 + 1)). (End)

A303817 Decimal expansion of 360/17.

Original entry on oeis.org

2, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5
Offset: 2

Views

Author

Omar E. Pol, Apr 30 2018

Keywords

Comments

Decimal expansion of the external angle of the regular heptadecagon (in degrees).
The repeating pattern [1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1] is the same as A007450. - Michael B. Porter, Jun 11 2018
Period 16. - Eric Chen, Jun 14 2018
Essentially the same as A021089. - R. J. Mathar, Aug 16 2018

Examples

			21.176470588235294117647058823529411764705882352941176470588235294...

Links

Crossrefs

Cf. A007450, A019434, A210644, A210649, A303816.

Programs

Formula

Equals 21 + (3/17) = 180 - A303816.
From Chai Wah Wu, Dec 20 2019: (Start)
a(n) = a(n-1) - a(n-8) + a(n-9) for n > 11.
G.f.: x^2*(-2*x^9 - 7*x^8 + 7*x^7 - 3*x^6 + 2*x^5 + x^4 - 6*x^3 + x - 2)/((x - 1)*(x^8 + 1)). (End)

A021308 Decimal expansion of 1/304.

Original entry on oeis.org

0, 0, 3, 2, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			0.00328947368421052631578947368421...

Links

Crossrefs

Cf. A007450, A021156.

Programs

  • Mathematica
    realDigitsRecip[304] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jul 18 2025 *)

Formula

From Chai Wah Wu, Apr 18 2024: (Start)
a(n) = a(n-1) - a(n-9) + a(n-10) for n > 13.
G.f.: x^2*(-5*x^11 + 3*x^10 + x^9 - 2*x^8 - 3*x^7 + 4*x^6 - 3*x^5 + 5*x^4 - x^3 - 6*x^2 + x - 3)/(x^10 - x^9 + x - 1). (End)

A370393 Decimal expansion of the area of a unit heptadecagon (17-gon).

Original entry on oeis.org

2, 2, 7, 3, 5, 4, 9, 1, 8, 9, 8, 4, 1, 6, 5, 5, 1, 4, 8, 2, 4, 2, 3, 7, 2, 3, 8, 7, 3, 9, 3, 7, 6, 3, 5, 7, 6, 1, 0, 6, 4, 1, 9, 9, 1, 4, 6, 9, 3, 3, 0, 9, 8, 8, 6, 0, 3, 5, 6, 5, 9, 4, 4, 0, 3, 9, 7, 2, 3, 2, 5, 1, 4, 8, 7, 9, 6, 7, 7, 7, 5, 7, 4, 7, 6, 4, 6
Offset: 2

Views

Author

Michal Paulovic, Feb 17 2024

Keywords

Comments

This constant multiplied by the square of the side length of a regular heptadecagon equals the area of that heptadecagon.
17^2 divided by this constant equals 68 * tan(Pi/17) = 12.71140300... which is the perimeter and the area of an equable heptadecagon with its side length 4 * tan(Pi/17) = 0.74772958... .
An equable rectangle with its perimeter and area = 17 has side lengths:
a = s^2/8 = (17 - sqrt(17)) / 4 = (17 - A010473) / 4 = 3.21922359...
b = 136/s^2 = (17 + sqrt(17)) / 4 = (17 + A010473) / 4 = 5.28077640...
where s is the parameter from the formula mentioned below.

Examples

			22.7354918984165514...

Links

Crossrefs

Cf. A007450, A010473, A019684 (Pi/17), A210644 (cos(2*Pi/17)), A210649, A228787, A241243, A329592, A343061.

Programs

  • Maple
    evalf(17 / (4 * tan(Pi/17)), 100);
  • Mathematica
    RealDigits[17 / (4 * Tan[Pi/17]), 10, 100][[1]]
  • PARI
    17 / (4 * tan(Pi/17))

Formula

Equals 17 / (4 * tan(Pi/17)) = 17 / (4 * A343061).
Equals 1 / (4 * A007450 * A343061).
Equals 17 * cos(Pi/17) / (4 * sin(Pi/17)).
Equals 17 * A210649 / (4 * A241243).
Equals 17 * A210649 / (2 * A228787).
Equals 17 * cot(Pi/17) / 4.
Equals 17 * sqrt(4 / (s^2 - 2 * s - 4 * sqrt(17 + 3 * sqrt(17) - s - sqrt(17) * 16/s)) - 1/16) where s = sqrt(34 - 2 * sqrt(17)) = 4 * A329592.
The minimal polynomial is 4294967296*x^16 - 3103113871360*x^14 + 510054948143104*x^12 - 28954726431195136*x^10 + 653743432704327680*x^8 - 6011468019822067712*x^6 + 20881180982314634240*x^4 - 21552361799603318912*x^2 + 2862423051509815793.
Showing 1-7 of 7 results.

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

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