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 1-5 of 5 results.

A096859 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the count of distinct terms arising in trajectory; a(n)=t(n)+c(n)=t+c, where t=number of transient terms, c=number of recurrent terms (in the terminal cycle).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 1, 2, 3, 3, 1, 3, 2, 2, 3, 3, 4, 2, 2, 4, 2, 2, 3, 4, 2, 4, 4, 2, 3, 4, 4, 4, 5, 4, 3, 5, 4, 4, 4, 2, 5, 3, 4, 4, 4, 4, 2, 4, 3, 4, 6, 5, 5, 4, 5, 5, 4, 4, 2, 4, 5, 3, 4, 4, 3, 5, 4, 5, 3, 4, 2, 4, 4, 3, 3, 5, 3, 5, 3, 4, 4, 4, 3, 4, 5, 5, 3, 4, 3, 3, 3, 5, 3, 5, 2, 6, 4, 3, 7, 5, 3, 3, 3, 5
Offset: 1

Views

Author

Labos Elemer, Jul 21 2004

Keywords

Examples

			n=255: list={255,144,360,288,[432,480],432,...}, t=transient=4, c=cycle=2, a(255)=t+c=6;
n=244: list={244,180,144,360,288,[432,480],432,...}, t=5, c=2, a(244)=7.

Links

Crossrefs

Cf. A000010, A000203, A062401, A062402, A095955, A096860, A096861, A096865.
Cf. also A096862.

Programs

  • Mathematica
    fs[x_] :=EulerPhi[DivisorSigma[1, x]] itef[x_, len_] :=NestList[fs, x, len] Table[Length[Union[itef[2^w, 20]]], {w, 1, 256}] (* len=20 at n<=256 is suitable *)
  • Scheme
    (define (A096859 n) (let loop ((visited (list n)) (i 1)) (let ((next (A062401 (car visited)))) (cond ((member next visited) i) (else (loop (cons next visited) (+ 1 i))))))) ;; Antti Karttunen, Nov 18 2017

A096865 Function A062401(x) = phi(sigma(x)) is iterated. Starting with n, a(n) is the smallest term arising in trajectory, either in transient or in terminal cycle.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 8, 9, 4, 4, 12, 4, 8, 8, 16, 4, 16, 8, 12, 16, 12, 8, 16, 16, 12, 16, 16, 8, 16, 16, 32, 16, 16, 16, 36, 16, 16, 16, 16, 12, 32, 12, 16, 16, 16, 16, 48, 36, 48, 16, 32, 16, 32, 16, 32, 32, 16, 16, 48, 16, 32, 48, 64, 16, 48, 32, 36, 32, 48, 16, 72, 36, 36, 48
Offset: 1

Views

Author

Labos Elemer, Jul 21 2004

Keywords

Examples

			n=255: list={255,144,360,288,[432,480],432,...}, a(255)=144 as a transient term;
n=254: list={254,[128],128,...}, a(254)=128, as a fixed point.

Links

Crossrefs

Cf. A062401, A062402, A095955, A096859, A096860, A096861 (largest term).
Cf. also A096866.

Programs

  • Mathematica
    fs[x_] :=EulerPhi[DivisorSigma[1, x]] itef[x_, hos_] :=NestList[fs, x, hos] Table[Min[itef[w, 20]], {w, 1, 256}]
  • Scheme
    (define (A096865 n) (let loop ((visited (list n)) (m n)) (let ((next (A062401 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (min m next))))))) ;; Antti Karttunen, Nov 18 2017

A096864 Function A062402(x) = sigma(phi(x)) is iterated. Starting with n, a(n) is the largest term arising in trajectory, either in transient or in terminal cycle.

Original entry on oeis.org

1, 2, 3, 4, 12, 6, 12, 12, 12, 12, 18, 12, 28, 14, 15, 16, 72, 18, 72, 20, 28, 22, 36, 24, 42, 28, 72, 28, 72, 30, 72, 72, 42, 72, 72, 36, 252, 72, 72, 72, 90, 42, 252, 44, 72, 46, 72, 72, 252, 50, 252, 72, 252, 72, 90, 72, 252, 72, 90, 72, 168, 72, 252, 252, 168, 66, 168, 252
Offset: 1

Views

Author

Labos Elemer, Jul 21 2004

Keywords

Examples

			n=256: list={256,255,255}, a(256)=256 as a transient term;
n=101: list={101,217,546,403,1170,819,[1240,1512],1240,...}, a(101)=1512 as a cycle term.

Links

Crossrefs

Cf. A062401, A062402, A066437, A096862, A096863, A096866 (smallest term), A096993.
Cf. also A096861.

Programs

  • Mathematica
    gf[x_] :=DivisorSigma[1, EulerPhi[x]] gite[x_, hos_] :=NestList[gf, x, hos] Table[Max[gite[w, 20]], {w, 1, 256}]
Table[Max[NestList[DivisorSigma[1,EulerPhi[#]]&,n,20]],{n,70}] (* Harvey P. Dale, May 13 2019 *)
  • Scheme
    (define (A096864 n) (let loop ((visited (list n)) (m n)) (let ((next (A062402 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (max m next))))))) ;; Antti Karttunen, Nov 18 2017

Formula

a(n) = max(n, A066437(n)). - Antti Karttunen, Dec 06 2017

A096860 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the count of distinct terms arising in the transient of this trajectory, that is: a(n) = A096859(n) - A095955(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 0, 2, 1, 1, 4, 2, 3, 1, 3, 3, 1, 1, 0, 1, 3, 1, 2, 1, 1, 3, 2, 3, 1, 1, 0, 2, 2, 1, 1, 3, 1, 3, 1, 2, 2, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 3, 1, 3, 0, 4, 2, 1, 5, 3, 1, 1, 1, 3
Offset: 1

Views

Author

Labos Elemer, Jul 21 2004

Keywords

Examples

			n=255: list={255,144,360,288,[432,480],432,...}, t=transient=4, c=cycle=2, a(255)=t=4;
n=244: list={244,180,144,360,288,[432,480],432,...}, a(244)=4.
a(n)=0 means that n is a recurrent term from A096850.

Links

Crossrefs

Cf. A000010, A000203, A062401, A062402, A095955, A096859, A096861, A096865.
Cf. also A096863.

Programs

  • Mathematica
    With[{nn = 120}, Array[Length@ Union@ # - Length@ Select[Tally@ #, Last@ # > 1 &] &@ NestList[EulerPhi@ DivisorSigma[1, #] &, #, nn] &, 105]] (* Michael De Vlieger, Nov 18 2017 *)
  • Scheme
    (define (A096860 n) (let loop ((visited (list n))) (let ((next (A062401 (car visited)))) (cond ((member next visited) => (lambda (transientplusone) (- (length transientplusone) 1))) (else (loop (cons next visited))))))) ;; Antti Karttunen, Nov 18 2017

A096863 Function A062402(x)=sigma(phi(x)) is iterated. Starting with n, a(n) is the count of transient terms of trajectory.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 0, 2, 1, 0, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 4, 2, 1, 2, 1, 1, 4, 2, 3, 1, 5, 2, 2, 1, 2, 2, 2, 0, 1, 1, 2, 3, 1, 2, 2, 3, 2, 1, 2, 0, 1, 2, 2, 2, 1, 1, 1, 3, 4, 2, 3, 1, 3, 4, 4, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 2, 6, 3, 2, 1, 1
Offset: 1

Views

Author

Labos Elemer, Jul 21 2004

Keywords

Comments

a(n)=0 means that n is a recurrent term from A096998.

Examples

			n=256: list={256,255,255}, a(256)=1;
n=101: list={101,217,546,403,1170,819,[1240,1512],1240,...,a(101)=6;

Crossrefs

Cf. A062401, A062402, A095955, A096859-A096866, A096993.

Formula

a(n) = A096861(n)-A096993(n).
Showing 1-5 of 5 results.

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