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.

A098189 Sum of unitary divisors minus Euler phi: a(n) = A034448(n) - A000010(n).

Original entry on oeis.org

0, 2, 2, 3, 2, 10, 2, 5, 4, 14, 2, 16, 2, 18, 16, 9, 2, 24, 2, 22, 20, 26, 2, 28, 6, 30, 10, 28, 2, 64, 2, 17, 28, 38, 24, 38, 2, 42, 32, 38, 2, 84, 2, 40, 36, 50, 2, 52, 8, 58, 40, 46, 2, 66, 32, 48, 44, 62, 2, 104, 2, 66, 44, 33, 36, 124, 2, 58, 52, 120, 2, 66, 2, 78, 64, 64, 36, 144, 2
Offset: 1

Views

Author

Labos Elemer, Sep 03 2004

Keywords

Examples

			a(1) = 1 - 1 = 0.

Links

Crossrefs

Cf. A034448, A000010, A063919, A098190, A098191, A098192, A098193, A098194, A098195.

Programs

  • Mathematica
    Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &] - EulerPhi@ n, {n, 120}] (* Michael De Vlieger, Mar 01 2017 *)
  • PARI
    a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - eulerphi(n); \\ Michel Marcus, Feb 25 2014
  • PARI
    a(n)=my(f=factor(n)); prod(k=1, #f[, 2], f[k, 1]^f[k, 2]+1) - eulerphi(f) \\ Charles R Greathouse IV, Mar 01 2017

Formula

a(n) > A063919(n) if n > 1.
a(A000040(k)) = 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/(12*zeta(3)) - 3/Pi^2 = 0.380252... . - Amiram Eldar, Aug 21 2023

Extensions

Edited by R. J. Mathar, Mar 02 2009

A098195 Starting values x such that the map x -> A098189(x) enters any cycle of length 29.

Original entry on oeis.org

246, 250, 274, 276, 278, 282, 345, 356, 382, 386, 390, 392, 399, 400, 405, 424, 438, 468, 474, 478, 482, 484, 486, 490, 510, 522, 524, 534, 556, 562, 566, 570, 578, 579, 591, 594, 598, 602, 614, 618, 620, 621, 622, 626, 628, 630, 642, 645, 648, 650, 662
Offset: 1

Views

Author

Labos Elemer, Sep 03 2004

Keywords

Comments

Iterating the map x -> A098189(x) may enter a cycle with 29 members (and there may be distinct cycles each with 29 members). The sequence lists all starting values of x such that (after some transient x) one of these cycles of length 29 is entered.
See other attractors and basins of attracted terms in A098191-A098195.
Corresponding number of transient terms for each term in a(n): {26, 25, 25, 25, 24, 23, 25, 26, 23, 22, 21, 24, 26, 24, 26, 25, 22, 27, 39, 17, 16, 22, 15, 27, 22, 27, 25, 25, 26, 16, 15, 14, 23, 25, 25, 33, 22, 39, 14, 13, 34, 26, 16, 15, 18, 14, 23, 34, 28, 20, 23, ...}. - Michael De Vlieger, Mar 01 2017

Examples

			282 is in the sequence since iterating the map x -> A098189(x) on that number yields 23 transient terms {282, 484, 390, 912, 1072, 628, 478, 482, 486, 570, 1296, 962, 1164, 1576, 998, 1002, 1684, 1270, 1800, 1860, 3360, 5568, 6008} then enters a cycle of 29 terms {3768, 4440, 7056, 6484, 4870, 6840, 9072, 8560, 7624, 4778, 4782, 7984, 4516, 3394, 3398, 3402, 4884, 7680, 10264, 6428, 4828, 4240, 3844, 2950, 3520, 3400, 2932, 2206, 2210}. - _Michael De Vlieger_, Mar 01 2017

Links

Crossrefs

Cf. A063919, A098189, A098190, A098191, A098192, A098193, A098194.

Programs

  • Mathematica
    Lookup[#, 29] &@ PositionIndex@ #[[All, -1]] &@ Table[If[n == 1, {0, 1}, Function[s, Function[t, {#, First@ Differences@ Take[Flatten@ t[[# + 1]], 2]} &@ Count[DeleteDuplicates@ t, k_ /; Length@ k == 1]]@ Map[Position[s, #] &, s]]@ NestList[Function[n, DivisorSum[n, # &, CoprimeQ[#, n/#] &] - EulerPhi@ n], n, n + 120]], {n, 800}] (* Michael De Vlieger, Mar 01 2017, Version 10 *)

Formula

{x: A098190(x) = 29}.

Extensions

Edited by R. J. Mathar, May 15 2009

A098192 Terms in a specific cycle of length 29 of the map x->A098189(x).

Original entry on oeis.org

2206, 2210, 2932, 2950, 3394, 3398, 3400, 3402, 3520, 3768, 3844, 4240, 4440, 4516, 4778, 4782, 4828, 4870, 4884, 6428, 6484, 6840, 7056, 7624, 7680, 7984, 8560, 9072, 10264
Offset: 1

Views

Author

Labos Elemer, Sep 03 2004

Keywords

Comments

Iteration of the map x -> A098189(x) enters cycles of various lengths.
The 29 terms in the first cycle where A098190(x)=29 are listed here in ascending order; the example section of A098190 shows them in the mapping order.
See other attractors in A098191-A098195.

Examples

			The cycle follows the mapping 3768 -> 4440 -> ... -> 2210-> 3768 (returning to the first element).
This cycle is entered, for example, if the mapping is started at x=246 (Cf. A098190).

Crossrefs

Cf. A063919, A098189, A098190, A098191, A098193, A098194, A098195.

Programs

  • Mathematica
    Function[s, Union@ Drop[s, #] &@ Count[DeleteDuplicates@ Map[Position[s, #] &, s], k_ /; Length@ k == 1]]@ NestList[Function[n, DivisorSum[n, # &, CoprimeQ[#, n/#] &] - EulerPhi@n], 246, 10^3] (* Michael De Vlieger, Mar 01 2017 *)
  • PARI
    A098189(n)=my(f=factor(n)); prod(k=1,#f~, f[k,1]^f[k,2]+1) - eulerphi(f)
a(n)=if(n>1, A098189(n-1), 2206) \\ Charles R Greathouse IV, Mar 01 2017

Extensions

Edited by R. J. Mathar, May 15 2009

A098193 Terms in a specific cycle of length 7 of the map x->A098189(x).

Original entry on oeis.org

1608, 1748, 1920, 2028, 2088, 2584, 2776
Offset: 1

Views

Author

Labos Elemer, Sep 03 2004

Keywords

Comments

The map enters a cycle of length 7 if started at x=294, indicated by A098190(294)=7.
The group members of this cycle are listed here in ascending order.
See other attractors in A098191-A098195.

Examples

			An iteration started at 294 leads to a attractor which cycles through 7 numbers:
294->516->712->458->462->1032->1248->1464->1752->[2088->2028->2776->1748->1608->1920->2584->2088,..].
After 9 transients, the cycle (indicated by bracketing its members) is entered.

Crossrefs

Cf. A063919, A098189, A098190, A098191, A098192, A098194, A098195.

Programs

Extensions

Edited by R. J. Mathar, May 15 2009

A098194 Starting values x such that the map x -> A098189(x) enters any cycle of length 53.

Original entry on oeis.org

46, 48, 50, 52, 54, 56, 58, 62, 66, 68, 69, 72, 82, 86, 88, 90, 93, 95, 96, 100, 106, 110, 112, 115, 117, 119, 123, 124, 132, 133, 135, 140, 141, 143, 145, 154, 155, 156, 158, 159, 162, 166, 168, 170, 174, 175, 176, 177, 180, 186, 187, 192, 195, 196, 200, 201
Offset: 1

Views

Author

Labos Elemer, Sep 03 2004

Keywords

Comments

Iterating the map x -> A098189(x) may enter a cycle with 53 members
(and there may be distinct cycles each with 53 members). The sequence
lists all starting values of x such that (after some transient x) one of these cycles of length 53 is entered.

Examples

			None of the values of A098192, A098193 or A098195 is in the sequence, because these will enter or are already in cycles of length 29 or 7.

Crossrefs

Cf. A063919, A098189, A098190, A098191, A098192, A098193, A098195.

Formula

{x: A098190(x) = 53}.

Extensions

Edited by R. J. Mathar, May 15 2009
Showing 1-5 of 5 results.

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