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.

A097026 Function f(x) = phi(x) + floor(x/2) is iterated, starting at x=n; a(n) is the length of terminal cycle (or 0 if no finite cycle exists).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1, 1, 2, 1, 4, 4, 2, 1, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 6, 4, 2, 2, 6, 4, 6, 2, 1, 2, 6, 2, 6, 2, 1, 1, 6, 2, 6, 6, 6, 1, 2, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6
Offset: 1

Views

Author

Labos Elemer, Aug 27 2004

Keywords

Comments

While iteration of phi(x) always leads to a fixed point, f(x) = phi(x) + incr(x) may result in cycles or be divergent. What is the magnitude of the added incrementing function?
Some initial values are hard to analyze. The first is n=163, so perhaps a(163)=0 by definition.
Observation regarding the above comment: most n <= 1000 have 1 <= a(n) <= 12; the following have unresolved cycles at 10^3 iterations of f(x): {163, 182, 196, 243, 283, 331, 423, 487, 495, 503, 511, 523, 533, 551, 559, 571, 583, 591, 593, ...}. - Michael De Vlieger, May 16 2017

Examples

			n=70: iteration list = {70, 59, 87, 99, 109, 162, 135, 139, 207, 235, 301, 402, 333, 382, 381, 442, [413, 554, 553, 744, 612, 498], 413}, a(70)=6;

Links

Crossrefs

Cf. A000010, A097027, A097028, A097029.

Programs

  • Mathematica
    With[{nn = 10^3}, Table[Count[Values@ PositionIndex@ NestList[EulerPhi@ # + Floor[#/2] &, n, nn], s_ /; Length@ s > 1], {n, 105}]] (* Michael De Vlieger, May 16 2017 *)
  • PARI
    findpos(newn, v) = {forstep(k=#v, 1, -1, if (v[k] == newn, return(k)););}
a(n) = {ok = 0; v = [n]; while(!ok, newn = eulerphi(n) + n\2; ipos = findpos(newn, v); if (ipos, ok = 1; break); v = concat(v, newn); n = newn;); #v - ipos + 1;} \\ Michel Marcus, Jan 03 2017

Formula

For n=2^j: a(2^j)=1, powers of 2 are fixed points.

A097028 Function f(x) = EulerPhi(x) + floor(x/2) is iterated; a(n) is the length of transient part and terminal cycle if the iteration was initiated at n. So a(n) is the number of distinct terms arising during iteration.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 3, 3, 4, 1, 1, 5, 2, 5, 3, 2, 2, 4, 4, 2, 3, 4, 4, 6, 4, 3, 1, 4, 5, 5, 4, 4, 5, 23, 5, 4, 5, 22, 6, 2, 2, 24, 6, 25, 3, 3, 4, 23, 3, 21, 5, 2, 3, 21, 3, 25, 26, 21, 1, 6, 24, 20, 25, 23, 22, 27, 4, 26, 27, 36, 28, 35, 22, 33, 5, 30, 31, 20, 25, 28, 29, 20, 26, 29
Offset: 1

Views

Author

Labos Elemer, Aug 27 2004

Keywords

Comments

Observation (see also Labos Elemer's comment at A097026): most n < 1000 have 1 <= a(n) <= 108; the following have a(n) > 1000 if finite: {163, 182, 196, 243, 283, 331, 423, 487, 495, 503, 511, 523, 533, 551, 559, 571, 583, 591, 593, 606, 611, 623, 642, 646, 651, 679, 685, 687, 725, 726, 729, 731, 732, 745, 746, 753, 755, 757, 758, 767, 779, 781, 783, 791, 799, 809, 811, 814, 839, 850, 855, 857, 859, 867, 869, 871, 875, 876, 885, 886, 888, 891, 895, 906, 908, 911, 913, 914, 915, 916, 921, 922, 923, 931, 937, 942, 959, 962, 964, 970, 971, 977, 985, 991}. - Michael De Vlieger, Feb 27 2017

Examples

			For n=70, iteration list = {70, 59, 87, 99, 109, 162, 135, 139, 207, 235, 301, 402, 333, 382, 381, 442, [413, 554, 553, 744, 612, 498], 413}, a(70) = 22.
n=2^j: a(2^j)=1, powers of 2 are fixed points, free of transients, so t + c = 0 + 1 = 1.

Crossrefs

Cf. A000010, A097026, A097028, A097029.

Programs

  • Mathematica
    Table[Length@ Union@ NestList[EulerPhi@ # + Floor[#/2] &, n, 10^3], {n, 10^3}] (* Michael De Vlieger, Feb 27 2017 *)

Formula

a(n) = A097026(n) + A097027(n) = c(n) + t(n).

A097027 Function f(x) = phi(x) + floor(x/2) is iterated; a(n) is the length of transient if the iteration was initiated at n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 3, 0, 0, 4, 1, 3, 2, 0, 0, 0, 3, 1, 2, 2, 3, 2, 0, 1, 0, 0, 1, 3, 0, 2, 3, 17, 1, 2, 1, 16, 2, 0, 0, 18, 2, 19, 1, 2, 2, 17, 1, 15, 3, 1, 2, 15, 1, 19, 20, 15, 0, 4, 18, 14, 19, 17, 16, 21, 2, 20, 21, 30, 22, 29, 16, 27, 3, 24, 25, 14, 19, 22, 23, 14, 20, 23
Offset: 1

Views

Author

Labos Elemer, Aug 27 2004

Keywords

Comments

Infinite iterations cannot be excluded, when a(n) is infinite. First at n=163?
For 1 <= n <= 1000, potentially infinite iterations at {163, 182, 196, 243, 283, 331, 423, 487, 495, 503, 511, 523, 533, 551, 559, 571, 583, 591, 593, 606, 611, 623, 642, 646, 651, 679, 685, 687, 725, 726, 729, 731, 732, 745, 746, 753, 755, 757, 758, 767, 779, 781, 783, 791, 799, 809, 811, 814, 839, 850, 855, 857, 859, 867, 869, 871, 875, 876, 885, 886, 888, 891, 895, 906, 908, 911, 913, 914, 915, 916, 921, 922, 923, 931, 937, 942, 959, 962, 964, 970, 971, 977, 985, 991} (tested to 1000 iterations). The maximum number of finite iterations in this range appears to be 96. - Michael De Vlieger, Mar 26 2017

Examples

			n=70: iteration list = [70, 59, 87, 99, 109, 162, 135, 139, 207, 235, 301, 402, 333, 382, 381, 442, [413, 554, 553, 744, 612, 498], 413], so a(70)=16.
n=2^j: a(2^j)=0, powers of 2 are fixed points of f, free of transients.

Crossrefs

Cf. A000010, A097026, A097028, A097029.

Programs

  • Maple
    a:= proc(n) local i, m, p;
      p:= proc() -1 end; forget(p);
      p(n):= 0; m:= n;
      for i do m:= numtheory[phi](m)+iquo(m, 2);
               if p(m)>-1 then return p(m) fi;
               p(m):= i
      od
    end:
seq(a(n), n=1..162);  # Alois P. Heinz, Nov 13 2015
  • Mathematica
    Table[Count[Values@ PositionIndex@ NestList[EulerPhi@ # + Floor[#/2] &, n, 10^3], k_ /; Length@ k == 1], {n, 89}] (* Michael De Vlieger, Mar 26 2017, Version 10 *)

A280577 a(n) = eulerphi(n) + floor(n/2).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 9, 8, 10, 9, 15, 10, 18, 13, 15, 16, 24, 15, 27, 18, 22, 21, 33, 20, 32, 25, 31, 26, 42, 23, 45, 32, 36, 33, 41, 30, 54, 37, 43, 36, 60, 33, 63, 42, 46, 45, 69, 40, 66, 45, 57, 50, 78, 45, 67, 52, 64, 57, 87, 46, 90, 61, 67, 64, 80, 53, 99, 66, 78
Offset: 1

Views

Author

Michel Marcus, Jan 05 2017

Keywords

Crossrefs

Cf. A000010, A004526.
Cf. A097026, A097027, A097028, A097029.

Programs

  • Maple
    with(numtheory): A280577:=n->phi(n)+floor(n/2): seq(A280577(n), n=1..100); # Wesley Ivan Hurt, Jan 05 2017
  • Mathematica
    Table[EulerPhi[n] + Floor[n/2], {n, 100}] (* Wesley Ivan Hurt, Jan 05 2017 *)
  • PARI
    a(n) = eulerphi(n) + floor(n/2);

Formula

a(n) = A000010(n) + A004526(n).
G.f.: x^2/((1 + x)*(1 - x)^2) + Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Jan 05 2017

A280578 Integers that are fixed points of some iteration of f(x) = phi(x) + floor(x/2), see A280577.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 21, 22, 23, 30, 32, 33, 36, 45, 46, 64, 128, 255, 256, 413, 498, 512, 513, 514, 553, 554, 580, 612, 744, 765, 766, 1024, 1073, 1125, 1162, 1250, 1540, 1544, 2048, 2241, 2413, 2457, 2458, 2522, 2524, 2596, 2754, 2889, 3348, 3352, 3474, 4096
Offset: 1

Views

Author

Michel Marcus, Jan 05 2017

Keywords

Comments

A097029 is the subsequence of integers that are fixed points of one iteration.
Up to 10^9, one can find cycles of length: 1, 2, 3, 4, 5, 6, 8, 10, 12, 20.

Links

Crossrefs

Cf. A000010, A004526.
Cf. A097026, A097027, A097028, A097029.
Cf. A280577.

Programs

  • PARI
    iscycle(v, nextn) = for (i=1, #v, if (v[i] == nextn, return (1); ); ); return (0);
fcycle(n, known) = {v = vector(1); v[1] = n; first = n; while ((nextn = eulerphi(n) + n\2) <= first, if (vecsearch(known, nextn), return([])); if (iscycle(v, nextn), return (v)); v = concat(v, nextn); n = nextn; ); return ([]);}
lista(nn) = {known = []; for (n = 1, nn, v = fcycle(n, known); if (#v, known = vecsort(concat(known, v)));); print(known);} \\ corrected by Michel Marcus, Mar 15 2022
Showing 1-5 of 5 results.

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