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.

A062401 a(n) = phi(sigma(n)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 4, 8, 12, 6, 4, 12, 6, 8, 8, 30, 6, 24, 8, 12, 16, 12, 8, 16, 30, 12, 16, 24, 8, 24, 16, 36, 16, 18, 16, 72, 18, 16, 24, 24, 12, 32, 20, 24, 24, 24, 16, 60, 36, 60, 24, 42, 18, 32, 24, 32, 32, 24, 16, 48, 30, 32, 48, 126, 24, 48, 32, 36, 32, 48, 24, 96, 36, 36, 60
Offset: 1

Views

Author

Jason Earls, Jul 08 2001

Keywords

Examples

			a(9) = 12 because sigma(9) = 13 and phi(13) = 12.

References

  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 14.

Links

Crossrefs

Cf. A000203, A000010, A062402, A096852, A096857, A096994, A096995, A033632.
Cf. A001229 (fixed points).

Programs

  • Haskell
    a062401 = a000010 . a000203  -- Reinhard Zumkeller, Jan 04 2013
  • Maple
    with(numtheory); A062401:=n->phi(sigma(n)); seq(A062401(n), n=1..50); # Wesley Ivan Hurt, Apr 07 2014
  • Mathematica
    Table[EulerPhi[DivisorSigma[1, n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)
  • PARI
    vector(150, n, eulerphi(sigma(n)))
  • PARI
    for (n=1, 10000, write("b062401.txt", n, " ", eulerphi(sigma(n))) ) \\ Harry J. Smith, Aug 07 2009

Formula

sigma(a(n)) = A062402(sigma(n)) or phi(A062402(n)) = a(phi(n)). - Labos Elemer, Jul 22 2004

A062402 a(n) = sigma(phi(n)).

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 12, 7, 12, 7, 18, 7, 28, 12, 15, 15, 31, 12, 39, 15, 28, 18, 36, 15, 42, 28, 39, 28, 56, 15, 72, 31, 42, 31, 60, 28, 91, 39, 60, 31, 90, 28, 96, 42, 60, 36, 72, 31, 96, 42, 63, 60, 98, 39, 90, 60, 91, 56, 90, 31, 168, 72, 91, 63, 124, 42, 144, 63, 84, 60, 144
Offset: 1

Views

Author

Jason Earls, Jul 08 2001

Keywords

Comments

Makowski and Schinzel conjectured in 1964 that a(n) = sigma(phi(n)) >= n/2 for all n (B42 of Guy Unsolved Problems...). This has been verified for various classes of numbers and proved to be true in general if it is true for squarefree integers (see Cohen's paper). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
Atanassov proves the above conjecture. - Charles R Greathouse IV, Dec 06 2016

Examples

			a(9)= 12 because phi(9)= 6 and sigma(6)= 12.

References

  • Krassimir T. Atanassov, One property of φ and σ functions, Bull. Number Theory Related Topics 13 (1989), pp. 29-37.
  • A. Makowski and A. Schinzel, On the functions phi(n) and sigma(n), Colloq. Math. 13, 95-99 (1964).
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 13.

Links

Crossrefs

Cf. A000203, A000010, A062401, A096852, A096857, A096994, A096995, A033632.

Programs

  • Haskell
    a062402 = a000203 . a000010  -- Reinhard Zumkeller, Jan 04 2013
  • Magma
    [SumOfDivisors(EulerPhi(n)): n in [1..100]] //  Marius A. Burtea, Jan 19 2019
  • Maple
    with(numtheory); A062402:=n->sigma(phi(n)); seq(A062402(k), k=1..100); # Wesley Ivan Hurt, Nov 01 2013
  • Mathematica
    Table[DivisorSigma[1, EulerPhi[n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)
  • PARI
    a(n)=sigma(eulerphi(n));
vector(150,n,a(n))
  • Python
    from sympy import divisor_sigma, totient
print([divisor_sigma(totient(n)) for n in range(1, 101)]) # Indranil Ghosh, Mar 18 2017

Formula

sigma(A062401(x)) = a(sigma(x)) or phi(a(x)) = A062401(phi(x)). - Labos Elemer, Jul 22 2004

A096994 Number of transient terms if f(x)=phi(sigma(x))=A062401 is iterated at initial value 2^n. Equilibrium terms are listed in A096852.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 4, 1, 2, 5, 14, 0, 5, 7, 2, 14, 8, 3, 64, 43, 81, 82, 76, 74, 47, 25, 42, 0
Offset: 0

Views

Author

Labos Elemer, Jul 22 2004

Keywords

Comments

For transient lengths for iterations of A062401(x) or A062402(x) if started at 2^n, A096994(n) + 1 = A096995(n). Corresponding cycle lengths satisfy A096852(n-1) = A096857(n). Behind these observations several relationships stand, e.g., sigma(A062401(x)) = A062402(sigma(x)) or phi(A062402(x)) = A062401(phi(x)).

Examples

			n=0: trajectory = {1,1,..} so a(0)=0;
n=14: transient-length=14, cycle-length=2, a(14)=14, A096852(14)=2; trajectory ={16384, 27000, 23040, 21600, 17280, 15360, 15488, 13824, 9600, 7680, 7200, 12960, 11880, 11520, [10368,14080], 10368, ...}.
Values of a(n) for n > 31, with -1 signifying transient lengths yet unknown after 10^4 iterations of f(x): -1, 7, 51, 70, 23, 39, 11, -1, 37, 107, 30, -1, 145, 25, 21, 36, -1, -1, -1, -1, 31, -1, 452, -1, 449, 447, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 40, -1, -1, -1, -1, -1, -1, -1, 279, -1. - _Michael De Vlieger_, May 15 2017

Crossrefs

Cf. A096852, A062401.

Programs

  • Mathematica
    With[{nn = 10^3}, Table[Count[Values@ PositionIndex@ #, k_ /; Length@ k == 1] &@ NestList[EulerPhi@ DivisorSigma[1, #] &, 2^n, nn] /. k_ /; k == nn + 1 -> -1, {n, 31}] ] (* Michael De Vlieger, May 15 2017, Version 10 *)

A096995 Number of transient terms if f(x) = sigma(phi(x)) = A062402 is iterated at initial value = 2^n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 3, 1, 2, 3, 5, 2, 3, 6, 15, 1, 6, 8, 3, 15, 9, 4, 65, 44, 82, 83, 77, 75, 48, 26, 43, 1
Offset: 0

Views

Author

Labos Elemer, Jul 22 2004

Keywords

Comments

For transient lengths of iterations A062401(x) or A062402(x), if started at 2^n, holds that A096994(n)+1 = a(n). Corresponding cycle lengths satisfy A096852(n-1) = A096857(n). Behind these observation several relationships stand, e.g., sigma(A062401(x)) = A062402(sigma(x)) or phi(A062402(x)) = A062401(phi(x)).
For initial value = 2^33 more than 38000 iterations did not lead to a recurrent term, so possibly there is no cycle. a(34) through a(39) are 8, 52, 71, 24, 40, 12. - Klaus Brockhaus, Jul 19 2007

Examples

			Trajectory of 2^0 is 1,1, ...; there are zero transient terms preceding the 1-cycle (1), so a(0) = 0.
Trajectory of 2^14 is 16384, 16383, 34200, 30480, 26520, 16380, 10200, 6138, 6045, 9906, 9920, 12264, 10200, ...; there are six transient terms preceding the 6-cycle (10200, 6138, 6045, 9906, 9920, 12264), so a(14) = 6.

Crossrefs

Cf. A062402, A062401, A096857, A096852, A096994.

Programs

Extensions

Edited and corrected by Klaus Brockhaus, Jul 19 2007

A097004 Function A062402(x)=phi(sigma(x)) is iterated. Starting with 2^n, the n-th power of 2, a(n) is the count of distinct terms arising in trajectory; a(n)=t(n)+c(n)=t+c, where t is the number of transient terms, c is the number of recurrent terms [in the terminal cycle].

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 5, 2, 4, 5, 11, 4, 4, 12, 17, 2, 8, 11, 14, 26, 11, 6, 80, 59, 100, 101, 95, 93, 60, 38, 55, 2
Offset: 0

Views

Author

Labos Elemer, Jul 21 2004

Keywords

Examples

			n=13: 2^n=8192, trajectory ={8192,8191,26208,[20440],.. }, a[13]=3+1=4 with 3 transients and one recurrent term.

Crossrefs

Cf. A000010, A000203, A062402, A096852, A096857.

Programs

  • Mathematica
    EulerPhi[DivisorSigma[1, x]] itef[x_, len_] :=NestList[fs, x, len] Table[Length[Union[itef[2^w, 1000]]], {w, 1, 256}]

A097002 A062402(x)=sigma(phi[x]) function is iterated; initial value=2^n; a(n)=smallest term of trajectory.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 6045, 24552, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823
Offset: 1

Views

Author

Labos Elemer, Jul 21 2004

Keywords

Examples

			n=13: 2^n=8192, trajectory ={8192, 8191, 26208, [20440], 20440, .. a[13]=8191 arose in transient.

Crossrefs

Cf. A000010, A000203, A062402, A096857.

Programs

  • Mathematica
    gf[x_] :=DivisorSigma[1, EulerPhi[x]] gite[x_, hos_] :=NestList[gf, x, hos] Table[Min[gite[2^w, 200]], {w, 1, 30}]
Showing 1-6 of 6 results.

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