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.

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

A266276 a(n) is the smallest number k such that phi(k) = n*phi(k-1).

Original entry on oeis.org

2, 3, 7, 1261, 11242771
Offset: 1

Views

Author

Jaroslav Krizek, Jan 26 2016

Keywords

Comments

a(n) >= A266269(n). - 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.

Crossrefs

Sequences of numbers n such that phi(n) = k*phi(n-1): {A001274 + 1} for k=1; A171271 = {A050472 + 1} for k=2; A266268 = {A067143 + 1} for k=3; A268126 = {A172314 + 1} for k=4; {A201253 + 1} for k=5.
Cf. A000010, A091439, A091456, A266269.

Programs

  • Magma
    a:=func; [a(n):n in[1..5]];
  • PARI
    a(n) = my(k=2, epk=1, enk); while ((enk=eulerphi(k)) != n*epk, epk = enk; k++); k; \\ Michel Marcus, Feb 20 2020

A091454 Integers k such that 3*phi(k) < k.

Original entry on oeis.org

30, 42, 60, 66, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 156, 168, 174, 180, 186, 198, 204, 210, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 294, 300, 306, 312, 318, 330, 336, 342, 348, 354, 360, 366, 372, 378, 390, 396, 402, 408, 414, 420
Offset: 1

Views

Author

Robert G. Wilson v, Jan 10 2004

Keywords

Comments

If k is present so are all m*k, m>1.
The smallest odd number in this sequence is 111546435 = 3*5*7*11*...*23. - T. D. Noe, Jan 15 2004

Links

Crossrefs

Cf. A000010 (phi), A054741, A066765, A091456.

Programs

  • Magma
    [k:k in [1..500]|3*EulerPhi(k) lt k]; // Marius A. Burtea, Jan 13 2020
  • Maple
    filter:= n -> 3*numtheory:-phi(n) < n:
select(filter, [$1..10000]); # Robert Israel, Jan 13 2020
  • Mathematica
    Select[ Range[437], 3EulerPhi[ # ] < # & ]
  • PARI
    is(n) = 3*eulerphi(n) < n; \\ Amiram Eldar, Apr 07 2024

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

A164347 The n-th term is the minimum number x such that x/Totient(x) >= n.

Original entry on oeis.org

2, 2, 6, 30, 210, 30030, 223092870, 13082761331670030, 3217644767340672907899084554130
Offset: 1

Views

Author

Fred Schneider, Aug 13 2009

Keywords

Comments

These numbers are all primorials. Primorials necessarily must be the minimum terms in this sequence (given the nature of Euler's Totient function).
Essentially the same as A091456. - R. J. Mathar, Aug 17 2009

Examples

			2 => 2/totient(2) = 2 (so it is both the first and 2nd entry of the sequence).
30 => 30/totient(30) = 15/4 >= 3.
210 => 210/totient(210) = 210/48 >= 4.

Links

Crossrefs

Each number n in this sequence is of the form: primorial(x). A164348, the related sequence, contains the x's.
Cf. A073087, A091456.

A355667 Least number phi(k) such that n * phi(k) < k, where phi is Euler's totient function.

Original entry on oeis.org

1, 2, 8, 48, 5760, 36495360, 1854081073152000, 400440702414394285778534400000, 165062110921422523175104166476600499887194872217600000000
Offset: 1

Views

Author

Nico Mexis, Jul 13 2022

Keywords

Comments

a(n) is the Euler totient of A091456(n). Useful for calculating a simple upper bound of the inverse phi function.

Examples

			If we know n=276, then all numbers k with phi(k)=n will satisfy k < 5*n since m=a(5) is the first number in the sequence with n < m.

Links

Crossrefs

Totients of A091456 or A091439.

Programs

  • CoCoA
    // CoCoA 5
define a(n) k := 1; p := 1; while k <= n*EulerTotient(k) do p := NextPrime(p); k := k*p; endwhile; return EulerTotient(k); enddefine;
  • Mathematica
    a[n_] := Module[{k = 1, p = 1, e}, While[k <= n*(e = EulerPhi[k]), p = NextPrime[p]; k *= p]; e]; Array[a, 9] (* Amiram Eldar, Jul 13 2022 *)
  • PARI
    a(n)=my(k=1); forprime(p=2, , if(n*eulerphi(k)

Formula

a(n) = A000010(A091456(n)).
a(n) = A000010(A091439(n+1)).
Showing 1-6 of 6 results.

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