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.

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

A362211 a(n) is the unique solution to A323410(x) = A362185(n).

Original entry on oeis.org

1, 6, 15, 21, 35, 11392, 1688, 10048, 53632, 101632, 5272, 2632, 6616, 50368, 1386, 102016, 1716, 1722, 161152, 4356, 11992, 92992, 4716, 101312, 589312, 2634, 644608, 3538, 3778, 898048, 30896, 16312, 3610, 3510, 4702, 1432576, 4626, 606976, 8908, 3738, 343936
Offset: 1

Views

Author

Amiram Eldar, Apr 11 2023

Keywords

Links

Crossrefs

Cf. A323410, A362185.
Similar sequences: A131826, A362212.

Programs

  • Mathematica
    ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 3000}, sol = solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, sol[[i]] = k; solnum[[i]]++], {k, 2, max^2}]; Join[{1}, sol[[Position[solnum, 1] // Flatten]]]]

Formula

A323410(a(n)) = A362185(n).

A362212 a(n) is the unique solution to A047994(x) = A361969(n).

Original entry on oeis.org

4, 8, 24, 16, 32, 76, 96, 64, 128, 184, 236, 216, 224, 316, 332, 384, 256, 344, 552, 428, 376, 424, 472, 556, 544, 768, 512, 692, 716, 608, 664, 796, 1128, 892, 908, 896, 1076, 864, 1416, 1132, 944, 1268, 1536, 1024, 1372, 1192, 1436, 1468, 1532, 1992, 1556, 1384
Offset: 1

Views

Author

Amiram Eldar, Apr 11 2023

Keywords

Comments

Are all the terms divisible by 4?

Links

Crossrefs

Cf. A047994, A135347, A361966, A361969.
Similar sequences: A131826, A362211.

Programs

  • Mathematica
    invUPhi[#][[1]]& /@ Select[Range[1250], Length[invUPhi[#]] == 1 &] (* using the function invUPhi from A361966 *)

Formula

a(n) = A135347(A361969(n)).
A047994(a(n)) = A361969(n).

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]

A362665 a(n) is the smaller of the two solutions to A091732(x) = A362664(n).

Original entry on oeis.org

1, 3, 4, 5, 11, 16, 33, 23, 29, 43, 69, 64, 47, 53, 76, 87, 59, 71, 79, 83, 141, 101, 103, 159, 107, 177, 131, 137, 213, 149, 163, 249, 167, 173, 236, 179, 235, 191, 197, 303, 309, 265, 321, 253, 223, 227, 229, 316, 239, 332, 251, 256, 393, 263, 269, 411, 283
Offset: 1

Views

Author

Amiram Eldar, Apr 29 2023

Keywords

Comments

The larger solution is 2*a(n).

Links

Crossrefs

Cf. A091732, A362484, A362664.
Similar sequences: A131826, A362211, A362212.

Programs

  • Mathematica
    invIPhi[#][[1]]& /@ Select[Range[300], Length[invIPhi[#]] == 2 &] (* using the function invIPhi from A362484 *)

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.

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