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 1-3 of 3 results.

A066663 Numbers n such that phi(n) is a proper substring of n.

Original entry on oeis.org

1320, 1640, 1768, 1996, 2640, 3960, 13200, 16400, 19984, 19996, 26400, 39600, 132000, 164000, 199996, 264000, 396000, 1320000, 1640000, 1999936, 2640000, 3960000, 13200000, 16400000, 16666240, 17999488, 18515584, 19999984, 19999996
Offset: 1

Views

Author

Vladeta Jovovic, Jan 11 2002

Keywords

Comments

Since the ratio n/phi(n) grows very slowly with n (see A091439), this sequence will be equal to A067206 (apart the initial 1) at least for all the terms smaller than 10^103. - Giovanni Resta, Apr 14 2017

Examples

			phi(1320)=320, phi(1640)=640, phi(1768)=768, phi(1996)=996, phi(2640)=640,..., phi(199999999984)=99999999984.

Links

Crossrefs

Cf. A000010, A036433, A091439, A067206.

Programs

  • Mathematica
    Select[Range[2, 10^5], StringPosition[ ToString@ #, ToString@ EulerPhi@ #] != {} &] (* Giovanni Resta, Apr 14 2017 *)

Extensions

Offset corrected and a(19)-a(29) from Donovan Johnson, May 08 2010

A093945 Primes of the form 5*10^n - 1.

Original entry on oeis.org

499, 4999, 49999, 4999999, 499999999999999, 4999999999999999999999999999999999999999999999999999999
Offset: 1

Views

Author

Rick L. Shepherd, Apr 17 2004

Keywords

Comments

Equivalently, primes of the form 4*10^n + 9*R_n, where R_n is the repunit (A002275) of length n.
If m is in the sequence then m appears at the end of m^3, in fact if n>1 and m=5*10^n-1 then m appears at the end of m^3. - Farideh Firoozbakht, Nov 10 2005
If n is in the sequence then 4n is a term of A067206. Namely the digits of 4n end in phi(4n) - the proof is easy. - Farideh Firoozbakht, Dec 30 2006
The next term -- a(7) -- has 211 digits. - Harvey P. Dale, Feb 20 2016

Links

Crossrefs

Cf. A056712 (corresponding n).
Cf. A067206.

Programs

  • Mathematica
    Select[Table[FromDigits[PadRight[{4},n,9]],{n,60}],PrimeQ] (* Harvey P. Dale, Feb 20 2016 *)

A248857 Composite numbers n such that n - phi(n) is a power of 10.

Original entry on oeis.org

1320, 1640, 1768, 1996, 13200, 16400, 19984, 19996, 132000, 164000, 199996, 1320000, 1640000, 1999936, 13200000, 16400000, 16666240, 17999488, 18515584, 19999984, 19999996, 132000000, 164000000, 164296960, 166662400, 199999936, 199999984, 1320000000
Offset: 1

Views

Author

Farideh Firoozbakht, Dec 31 2014

Keywords

Comments

Numbers n such that n - phi(n) is a positive power of 10.
Numbers n such that phi(n) = n - 10^floor(log(10,n)).
The most significant digit of all terms is equal to 1, since all terms are even and for even numbers n, phi(n) <= n/2.
If p = 5^(2n-1)*10^m-1 is prime then s = 4^n*p is in the sequence, because s - phi(s) = 10^(2n+m-1).
For n=1,2, ..., 6, ... smallest such term of the sequence respectively are 1996, 19984, 1999936, 1999999744, 19999999998976,19999999995904, ... .
Sequence A248858 gives number of digits of these terms.

Examples

			1320 is in the sequence because 1320 - phi(1320) = 10^3.

Links

Crossrefs

Cf. A000010, A067206, A066663, A244440, A248854, A248858.

Programs

  • Mathematica
    a248857[n_] := Select[Select[Range@n, CompositeQ[#] &], IntegerQ[Log10[# - EulerPhi[#]]] &]; a248857[10^7] (* Michael De Vlieger, Jan 07 2015 *)
  • PARI
    lista(nn) = forcomposite(n=2, nn, if (ispower(n-eulerphi(n),,&d) && (d==10), print1(n, ", "))); \\ Michel Marcus, Jan 06 2015

Extensions

a(22)-a(28) from Giovanni Resta, Apr 17 2017
Showing 1-3 of 3 results.

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

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