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 1-4 of 4 results.

A091439 Least k such that k/phi(k) >= n, where phi is Euler's totient function.

Original entry on oeis.org

1, 2, 6, 210, 30030, 223092870, 13082761331670030, 3217644767340672907899084554130, 1492182350939279320058875736615841068547583863326864530410, 16516447045902521732188973253623425320896207954043566485360902980990824644545340710198976591011245999110
Offset: 1

Views

Author

T. D. Noe, Jan 09 2004

Keywords

Links

Crossrefs

Subsequence of A002110.
Cf. A091456 (n * phi(k) < k), A256968, A266269, A266276.

Programs

  • Mathematica
    Table[prod=1; i=0; While[prod
  • PARI
    a(n) = my(p=1, i=0); while(pMichel Marcus, Oct 07 2019

Formula

a(n) = A002110(A256968(n)). - Michel Marcus, Oct 07 2019

A266268 Numbers n such that phi(n) = 3*phi(n-1).

Original entry on oeis.org

7, 13, 19, 37, 73, 91, 97, 109, 163, 193, 433, 487, 577, 703, 769, 793, 925, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 4699, 5551, 6697, 7999, 8701, 10369, 10591, 11803, 12289, 16471, 17497, 18433, 33251, 39367, 52489, 56791, 79249, 124357, 127927, 137899
Offset: 1

Views

Author

Jaroslav Krizek, Dec 26 2015

Keywords

Comments

Prime terms are in A058383.
See A266276(n) = the smallest numbers k such that phi(k) = n * phi(k-1) for n >=1: 2, 3, 7, 1261, 11242771, ...
Number of terms < 10^k: 1, 7, 17, 29, 41, 86, 205, 446, 1001, 2295, ..., . - Robert G. Wilson v, Jan 24 2016
All terms are == +-1 (mod 6) but mostly 1 (> 95%). - Robert G. Wilson v, Jan 24 2016

Examples

			19 is in the sequence because phi(19) = 18 = 3*phi(18) = 3*6.

Links

Crossrefs

Cf. A000010, A058383, A171271 (numbers n such that phi(n) = 2*phi(n-1)), A266276.

Programs

  • Magma
    [n: n in [2..2*10^5] | EulerPhi(n) eq 3*EulerPhi(n-1)]; // Vincenzo Librandi, Dec 26 2015
  • Mathematica
    Select[Range[5000], EulerPhi[ # ]==3*EulerPhi[ #-1]&] (* G. C. Greubel, Dec 26 2015 *)
  • PARI
    isok(n) = eulerphi(n) == 3*eulerphi(n-1); \\ Michel Marcus, Dec 27 2015
  • PARI
    lista(nn) = for(n=1, nn, if(eulerphi(n) == 3*eulerphi(n-1), print1(n, ", "))); \\ Altug Alkan, Jan 24 2016

Formula

a(n) = A067143(n) + 1.

A266269 a(n) is the smallest number k such that phi(k) >= n*phi(k-1).

Original entry on oeis.org

2, 3, 7, 211, 30031, 223092871, 13082761331670031, 3217644767340672907899084554131, 1492182350939279320058875736615841068547583863326864530411, 16516447045902521732188973253623425320896207954043566485360902980990824644545340710198976591011245999111
Offset: 1

Views

Author

Jaroslav Krizek, Jan 26 2016

Keywords

Comments

For the known terms, we have a(n) = 1 + A002110(A256968(n)) = 1 + A091439(n), which likely holds for most (if not all) terms overall. - Max Alekseyev, Jan 26 2025

Examples

			a(3) = 7 because 7 is the smallest number k such that phi(k) >= n*phi(k-1); phi(7) = 6 >= 3*phi(6) = 3*2.

Links

Crossrefs

Cf. A000010, A002110, A091439, A091456, A256968, A266276.

Programs

  • Magma
    a:=func; [a(n):n in[1..5]];
  • PARI
    a(n) = {my(k=2, e=1); while(n*e > e=eulerphi(k), k++); k; } \\ Jinyuan Wang, Nov 01 2020

Extensions

a(6)-a(8) from Jinyuan Wang, Nov 01 2020
a(9)-a(10) from Max Alekseyev, Jan 25 2025

A268126 Numbers n such that phi(n) = 4*phi(n-1).

Original entry on oeis.org

1261, 13651, 17557, 18721, 24511, 42121, 113611, 244531, 266071, 712081, 749911, 795691, 992251, 1080721, 1286731, 1458271, 1849471, 2271061, 2457691, 3295381, 3370771, 3414841, 3714751, 4061971, 4736491, 5314051, 5827081, 6566911, 6935083, 7303981, 7864081
Offset: 1

Views

Author

Jaroslav Krizek, Jan 26 2016

Keywords

Comments

See A266276(n) = the smallest numbers k such that phi(k) = n * phi(k-1) for n >=1: 2, 3, 7, 1261, 11242771, ...

Examples

			1261 is in the sequence because phi(1261) = 1152 = 4*phi(1260) = 4*288.

Crossrefs

Cf. A000010, A171271 (numbers n such that phi(n) = 2*phi(n-1)), A266268 (numbers n such that phi(n) = 3*phi(n-1)), A266276.
Cf. A256937.

Programs

  • Magma
    [n: n in [2..10^7] | EulerPhi(n) eq 4*EulerPhi(n-1)]
  • Mathematica
    Select[Range@10000000, EulerPhi@# == 4 EulerPhi[# - 1] &] (* Vincenzo Librandi, Jan 27 2016 *)
  • PARI
    isok(n) = (eulerphi(n) == 4*eulerphi(n-1)); \\ Michel Marcus, Jan 27 2016

Formula

a(n) = A172314(n) + 1. - Michel Marcus, Jan 27 2016
Showing 1-4 of 4 results.

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

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