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 41-47 of 47 results.

A283808 Numbers k such that phi(phi(k)) divides k, where phi(k) is A000010(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 32, 36, 40, 48, 54, 56, 64, 72, 80, 96, 108, 112, 128, 144, 160, 162, 192, 216, 224, 256, 288, 320, 324, 384, 432, 448, 486, 512, 576, 640, 648, 768, 864, 896, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1728, 1792, 1944, 2048, 2304, 2560
Offset: 1

Views

Author

Giovanni Resta, Mar 17 2017

Keywords

Comments

M. Hausman has proved (see Links) that a number belongs to this sequence if and only if it is of one of the following forms: 2^s, 2^s * 3^t, 5 * 2^t, or 7 * 2^t , where s >= 0 and t >= 1.

Examples

			56 is in the sequence because phi(phi(56)) = 8 divides 56.

Links

Crossrefs

Cf. A010554, A007694, A000010, A019278.

Programs

  • Mathematica
    Select[Range[1000], Mod[#, EulerPhi@ EulerPhi@ #] == 0 &]
  • PARI
    alias(e, eulerphi);
for(n = 1, 1000, if(!Mod(n,e(e(n))), print1(n,", "))) \\ Indranil Ghosh, Mar 18 2017
  • Python
    from sympy import totient as e
print([n for n in range(1, 1001) if n%e(e(n))==0]) # Indranil Ghosh, Mar 18 2017

Formula

Sum_{n>=1} 1/a(n) = 667/210. - Amiram Eldar, Dec 13 2024

A289125 Numbers n such that phi(n)/phi(phi(n)) > phi(m)/phi(phi(m)) for all m < n.

Original entry on oeis.org

1, 3, 7, 31, 211, 2311, 43891, 60653, 870871, 1023053, 13123111, 19417793, 300690391, 446235509, 6915878971, 12939711677, 200560490131
Offset: 1

Views

Author

Amiram Eldar, Jun 25 2017

Keywords

Comments

Erdős et al. proved that phi(n)/phi(phi(n)) is unbounded, thus this sequence is infinite.
A018239(k) = A002110(A014545(k)) + 1 is a term for k > 1. Are there terms m with omega(m) > 2? Is omega(phi(a(n + 1))) >= omega(phi(a(n)))? - David A. Corneth, Jun 28 2017

Links

Crossrefs

Cf. A000010, A001221, A002110, A010554, A014545, A018239.

Programs

  • Mathematica
    a = {}; k=1; rmax = 0; While[Length[a]<10,s = EulerPhi[ k]; s2 = EulerPhi[ s]; r = s/s2;  If[r > rmax, AppendTo[a, k]; rmax = r]; k++]; a
DeleteDuplicates[Table[{n,EulerPhi[n]/EulerPhi[EulerPhi[n]]},{n,11*10^5}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first ten terms of the sequence. *) (* Harvey P. Dale, Aug 17 2024 *)
  • PARI
    r=0; forfactored(n=1,10^10, t=eulerphi(n); t/=eulerphi(t); if(t>r, r=t; print1(n[1]", "))) \\ Charles R Greathouse IV, Jun 25 2017

Extensions

a(15)-a(17) from Giovanni Resta, Jul 01 2017

A380500 Table T(n,k) = phi(phi(prime(n)^k)), n >= 1, k >= 0, read by upwards antidiagonals, where phi = A000010.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 8, 6, 4, 1, 4, 12, 40, 18, 8, 1, 4, 40, 84, 200, 54, 16, 1, 8, 48, 440, 588, 1000, 162, 32, 1, 6, 128, 624, 4840, 4116, 5000, 486, 64, 1, 10, 108, 2176, 8112, 53240, 28812, 25000, 1458, 128, 1, 12, 220, 2052, 36992, 105456, 585640, 201684, 125000, 4374, 256
Offset: 1

Views

Author

Michael De Vlieger, Feb 04 2025

Keywords

Comments

For n >= 2, k >= 1, T(n,k) is the number of primitive roots of prime(n)^k.

Examples

			Table begins as follows:
n\k  0   1     2      3       4        5          6           7
---------------------------------------------------------------
1:   1   1     1      2       4        8         16          32
2:   1   1     2      6      18       54        162         486
3:   1   2     8     40     200     1000       5000       25000
4:   1   2    12     84     588     4116      28812      201684
5:   1   4    40    440    4840    53240     585640     6442040
6:   1   4    48    624    8112   105456    1370928    17822064
7:   1   8   128   2176   36992   628864   10690688   181741696

Links

Crossrefs

Cf. A000010, A000040, A001248, A008330, A010554, A046144, A104039, A246655.

Programs

  • Mathematica
    Table[EulerPhi[EulerPhi[Prime[#]^k]] &[n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten

Formula

T(n,k) = A010554(prime(n)^k) = A046144(prime(n)^k).
T(n,0) = 1.
T(n,1) = phi(prime(n)-1) = A008330(n).
T(n,2) = (prime(n)-1) * phi(prime(n)-1)
= (prime(n)-1)^2 * Product_{q|(prime(n)-1)} 1-1/q, prime q.
= A104039(n).
For k > 1, T(n,k) = prime(n)^(k-2) * A104039(n).
T(n,2) > prime(n) for n > 2.
T(n,k) < prime(n)^k for all n and for k > 0.

A075863 Numbers k such that phi(phi(k)) = sum of prime factors of k.

Original entry on oeis.org

8, 45, 55, 95, 126, 132, 198, 215, 228, 516, 2855, 6852, 7655, 18372, 276455, 663492
Offset: 1

Views

Author

Joseph L. Pe, Oct 15 2002

Keywords

Comments

a(17) > 2*10^9. - Hiroaki Yamanouchi, Sep 19 2014

Examples

			phi(phi(126)) = phi(36) = 12 and the sum of the prime factors of 126 = 2 * 3^2 * 7 is 2 + 3 + 7 = 12. Hence 126 belongs to the sequence.

Crossrefs

Cf. A008472, A010554.

Programs

  • Mathematica
    Select[Range[2, 10^5], EulerPhi[EulerPhi[ # ]] == Apply[Plus, Transpose[FactorInteger[ # ]][[1]]] &]
  • Python
    from sympy import primefactors, totient as phi
def ok(n): return n and phi(phi(n)) == sum(primefactors(n))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jun 19 2023

Extensions

a(15)-a(16) from Hiroaki Yamanouchi, Sep 19 2014

A293714 Numbers k such that phi(phi(k))/k < phi(phi(m))/m for all m < k, where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 6, 14, 18, 42, 90, 186, 198, 210, 450, 462, 930, 990, 1050, 2310, 7350, 14910, 16170, 66990, 169050, 177870, 371910, 1540770, 2312310, 4626930, 4834830, 19711230, 20030010, 30060030, 60150090, 62852790, 256245990, 260390130, 1061590530, 1068497430
Offset: 1

Views

Author

Amiram Eldar, Oct 15 2017

Keywords

Crossrefs

Cf. A000010, A010554.

Programs

  • Mathematica
    f[n_] := EulerPhi[EulerPhi[n]]/n; a={}; fm=10^10; Do[f1=f[k]; If[f1
  • PARI
    lista(nn) = {my(rmin = 2); for (n=1, nn, if ((r=eulerphi(eulerphi(n))/n) < rmin, rmin = r; print1(n, ", ")););}

Extensions

a(26)-a(34) from Michel Marcus, Oct 18 2017
a(35)-a(38) from Amiram Eldar, Nov 12 2024

A349867 Generalized multiplicative Euler phi function of order 2 (see El Hindi et al. link).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 4, 2, 2, 8, 8, 2, 6, 2, 2, 4, 10, 1, 8, 4, 6, 2, 12, 2, 8, 16, 4, 8, 4, 2, 12, 6, 4, 2, 16, 2, 12, 4, 4, 10, 22, 8, 12, 8, 8, 4, 24, 6, 8, 2, 6, 12, 28, 2, 16, 8, 4, 32, 8, 4, 20, 8, 10, 4, 24, 2, 24, 12, 8, 6, 8, 4, 24, 16
Offset: 1

Views

Author

Michel Marcus, Dec 03 2021

Keywords

Links

Crossrefs

Cf. A000010, A010554.

Programs

  • PARI
    phig(n, k) = {my(f=factor(n)); if (k==1, return(eulerphi(f))); for (j=1, #f~, my(p=f[j,1], alfa=f[j,2]); if (p % 2, f[j,1] = if (alfa < k, phig(p-1, k-1)*prod(i=k-alfa+1, k-1, phig(p, i)), phig(p-1, k-1)*p^(alfa-k)*prod(i=1, k-1, phig(p, i))), f[j,1] = if (alfa < 2*k, 1, 2^(alfa-1));); f[j,2] = 1;); factorback(f);}
a(n) = phig(n, 2);

A377402 Least k such that the ratio of the number of residues mod k coprime to k and the number of primitive roots mod k is greater than or equal to n for k such that at least one primitive root mod k exists. Equivalently, k such that floor(phi(k)/phi(phi(k)) is a record value for those k belonging to A033948.

Original entry on oeis.org

1, 3, 7, 211, 43891, 300690391
Offset: 1

Views

Author

Miles Englezou, Oct 26 2024

Keywords

Comments

These are the numbers k with the least proportion of primitive roots mod k to residues mod k coprime to k, of all m < k.
Let U(k) be the group of units of the ring Z/kZ, and Gen(k) the set of distinct single element generators of U(k). Then a(n) is equivalently those k for which floor(|U(k)|/|Gen(k)|) is a record value for those k with cyclic U(k).
Some data:
---------------------------------------------------------------------------------
n a(n) log(a(n)) phi(a(n)) max prime(r)#|phi(a(n)) r
---------------------------------------------------------------------------------
1 1 0 1 0 0
2 3 1.10... 2 2 1
3 7 1.96... 2*3 6 2
4 211 5.35... 2*3*5*7 210 4
5 43891 10.69... 2*3*5*7*11*19 2310 5
6 300690391 19.52... 2*3*5*7*11*13*17*19*31 9699690 8
---------------------------------------------------------------------------------
For n > 1, is a(n) necessarily prime? Furthermore, is a(n) necessarily a prime such that phi(a(n)) is squarefree? Lastly, is every phi(a(n)) divisible by a maximal primorial prime(r)# of length r <= omega(phi(a(n))), such that if a maximal prime(s)# divides phi(a(m)) and n < m, then r < s?
Based on the behavior of log(a(n)), we may expect a(7) to be found in the vicinity of floor(exp(40)) = 235385266837019985.

Examples

			There are 2 residues mod 3 coprime to 3, and only 1 is a primitive root. 3 is the least k for which the floor of the ratio is 2, and so a(2) = 3.
There are 210 residues mod 211 coprime to 211, and 48 are primitive roots. Floor(210/48) = 4, and 211 is the least k for which the floor of the ratio is 4, and so a(4) = 211.

Crossrefs

Cf. A000010, A010554, A033948, A046144.

Programs

  • PARI
    S=[1]; for(n=1,100000, if(#znstar(n).cyc>1,next); f=eulerphi; if(floor(f(n)/f(f(n)))>floor(f(S[length(S)])/f(f(S[length(S)]))), S=concat(S,n))); print(S)

Formula

Numbers k for which floor(A000010(A033948(k))/A046144(A033948(k)) is a record value.
Previous Showing 41-47 of 47 results.

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

Content is available under The OEIS End-User License Agreement.