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.

A083235 First differences of A063742, the possible values for cototients.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Labos Elemer, May 20 2003

Keywords

Comments

Differences between consecutive numbers in the range of A051853.

Examples

			First missing number in A063742 is 10=A005278[1], so a[10]=2 is the first difference here > 1.

Programs

  • Mathematica
    t0[x_] := Table[j, {j, 1, x}] t=Table[w-EulerPhi[w], {w, 1, 10000}]; u=Union[%]; Delete[u-RotateRight[u], 1]

Formula

a(n) = A063742(n+1) - A063742(n).

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 23 2007

A131826 a(n) is the unique solution to Cototient(x) = A131825(n).

Original entry on oeis.org

1, 4, 9, 25, 10, 26, 34, 38, 30, 52, 74, 76, 82, 122, 90, 134, 146, 148, 114, 130, 194, 202, 206, 170, 226, 244, 186, 268, 292, 228, 290, 386, 388, 398, 404, 412, 370, 482, 366, 488, 434, 514, 430, 584, 518, 578, 614, 626, 450, 674, 462, 580, 610, 746, 558, 772
Offset: 1

Views

Author

Franz Vrabec, Jul 20 2007

Keywords

Examples

			a(5) = 10 is the unique solution to Cototient(x) = A131825(5) = 6.

Links

Crossrefs

Cf. A051953, A063742, A131825.

Programs

  • PARI
    lista(nn) = {my(c, r, v=vector(nn^2+2, i, i - eulerphi(i))); for(k=0, nn, c=0; for(i=1, k*k+2, if(k==v[i], r=i; c++)); if(c==1, print1(r, ", "))); } \\ Jinyuan Wang, Mar 22 2020

Extensions

More terms from Jinyuan Wang, Mar 22 2020

A131825 Numbers k such that cototient(x) = k has exactly 1 solution.

Original entry on oeis.org

0, 2, 3, 5, 6, 14, 18, 20, 22, 28, 38, 40, 42, 62, 66, 68, 74, 76, 78, 82, 98, 102, 104, 106, 114, 124, 126, 136, 148, 156, 178, 194, 196, 200, 204, 208, 226, 242, 246, 248, 254, 258, 262, 296, 302, 306, 308, 314, 330, 338, 342, 356, 370, 374, 378, 388, 398, 400, 408, 416, 418, 422, 426, 434, 438
Offset: 1

Views

Author

Franz Vrabec, Jul 19 2007

Keywords

Examples

			6 = cototient(10) and for all x<>10, cototient(x) <> 6.

Links

Crossrefs

Cf. A051953, A063742, A131826.

Programs

  • Maple
    N:= 1000: # for all terms <= N
cotot:= n -> n - numtheory:-phi(n):
V:= Vector(N):
for n from 2 to N^2 do
  v:= cotot(n);
  if v  > N then next fi;
  V[v]:= V[v]+1;
od:
0, op(select(t -> V[t]=1, [$1..N])); # Robert Israel, May 26 2019
  • PARI
    lista(nn) = {my(v=vector(nn^2+2, i, i - eulerphi(i))); for(k=0, nn, if(sum(i=1, k*k+2, k==v[i])==1, print1(k, ", "))); } \\ Jinyuan Wang, Mar 21 2020

Extensions

More terms from Robert Israel, May 26 2019

A362213 Irregular table read by rows in which the n-th row consists of all the numbers m such that cototient(m) = n, where cototient is A051953.

Original entry on oeis.org

4, 9, 6, 8, 25, 10, 15, 49, 12, 14, 16, 21, 27, 35, 121, 18, 20, 22, 33, 169, 26, 39, 55, 24, 28, 32, 65, 77, 289, 34, 51, 91, 361, 38, 45, 57, 85, 30, 95, 119, 143, 529, 36, 40, 44, 46, 69, 125, 133, 63, 81, 115, 187, 52, 161, 209, 221, 841, 42, 50, 58, 87, 247, 961
Offset: 2

Views

Author

Amiram Eldar, Apr 11 2023

Keywords

Comments

The offset is 2 since cototient(p) = 1 for all primes p.
The 0th row consists of one term, 1, since 1 is the only solution to cototient(x) = 0.

Examples

			The table begins:
  n   n-th row
  --  -----------
   2  4;
   3  9;
   4  6, 8;
   5  25;
   6  10;
   7  15, 49;
   8  12, 14, 16;
   9  21, 27;
  10
  11  35, 121;
  12  18, 20, 22;

Links

Crossrefs

Cf. A005278, A051953, A063507, A063740 (row lengths), A063741, A063742, A063748, A100827, A101373, A131825, A131826.
Similar sequences: A032447, A361966, A362180.

Programs

  • Mathematica
    With[{max = 50}, cot = Table[n - EulerPhi[n], {n, 1, max^2}]; row[n_] := Position[cot, n] // Flatten; Table[row[n], {n, 2, max}] // Flatten]

A058817 Even cototient numbers.

Original entry on oeis.org

0, 2, 4, 6, 8, 12, 14, 16, 18, 20, 22, 24, 28, 30, 32, 36, 38, 40, 42, 44, 46, 48, 54, 56, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 88, 90, 92, 94, 96, 98, 102, 104, 106, 108, 110, 112, 114, 118, 120, 124, 126, 128, 132, 136, 138, 140, 142, 144, 148, 150, 152
Offset: 1

Views

Author

Labos Elemer, Jan 04 2001

Keywords

Examples

			88 is here because it is the cototient of 120: 88 = 120-phi(120) = 120-32 = 88.

Links

Crossrefs

Cf. A000010, A051953, A063742, A005277, A005278, A002202, A058763.

Programs

  • Mathematica
    With[{max = 300}, Union@ Select[Table[n - EulerPhi[n], {n, 1, max^2}], # < max && EvenQ[#] &]] (* Amiram Eldar, Jan 12 2024 *)

Formula

Even terms of A063742.

Extensions

Offset corrected by Donovan Johnson, Nov 17 2013
a(1) = 0 inserted by Amiram Eldar, Jan 12 2024

A131827 Numbers k such that cototient(x) = k has exactly 2 solutions.

Original entry on oeis.org

4, 7, 9, 11, 13, 15, 36, 37, 44, 46, 54, 56, 70, 80, 84, 88, 90, 92, 94, 112, 118, 138, 142, 152, 158, 160, 162, 164, 166, 174, 176, 182, 184, 188, 198, 210, 212, 214, 228, 230, 234, 236, 252, 272, 276, 278, 282, 304, 312, 316, 318, 320, 322, 328, 352, 354, 364
Offset: 1

Views

Author

Franz Vrabec, Jul 19 2007

Keywords

Examples

			4 = cototient(6) = cototient(8) and there are no other solutions.

Crossrefs

Cf. A051953, A063742, A131825, A131826.

Programs

  • PARI
    lista(nn) = {my(v=vector(nn^2, i, i - eulerphi(i))); for(k=0, nn, if(sum(i=1, k*k, k==v[i])==2, print1(k, ", "))); } \\ Jinyuan Wang, Mar 21 2020

Extensions

More terms from Jinyuan Wang, Mar 21 2020
Showing 1-6 of 6 results.

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

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