A131825 -id:A131825 - cp's OEIS frontend

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

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

Franz Vrabec, Jul 20 2007

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..15628

Cf. A051953, A063742, A131825.

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

More terms from Jinyuan Wang, Mar 22 2020

A063740 Number of integers k such that cototient(k) = n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 2, 0, 2, 3, 2, 1, 2, 3, 3, 1, 3, 1, 3, 1, 4, 4, 3, 0, 4, 1, 4, 3, 3, 4, 3, 0, 5, 2, 2, 1, 4, 1, 5, 1, 4, 2, 4, 2, 6, 5, 5, 0, 3, 0, 6, 2, 4, 2, 5, 0, 7, 4, 3, 1, 8, 4, 6, 1, 3, 1, 5, 2, 7, 3, 5, 1, 7, 1, 8, 1, 5, 2, 6, 1, 9, 2, 6, 0, 4, 2, 10, 2, 4, 2, 5, 2, 7, 5, 4, 1, 8, 0, 9, 1, 6, 1, 7

Offset: 2

Labos Elemer, Aug 13 2001

nonn

Note that a(0) is also well-defined to be 1 because the only solution to x - phi(x) = 0 is x = 1. - Jianing Song, Dec 25 2018

			Cototient(x) = 101 for x in {485, 1157, 1577, 1817, 2117, 2201, 2501, 2537, 10201}, with a(101) = 8 terms; e.g. 485 - phi(485) = 485 - 384 = 101. Cototient(x) = 102 only for x = 202 so a(102) = 1.

T. D. Noe, Table of n, a(n) for n = 2..10000

Cf. A000010, A051953, A063507.
Cf. A063748 (greatest solution to x-phi(x)=n).
Cf. A005278, A063741, A131825.

Table[Count[Range[n^2], k_ /; k - EulerPhi@ k == n], {n, 2, 105}] (* Michael De Vlieger, Mar 17 2017 *)

first(n)=my(v=vector(n),t); forcomposite(k=4,n^2, t=k-eulerphi(k); if(t<=n, v[t]++)); v[2..n] \\ Charles R Greathouse IV, Mar 17 2017

From Amiram Eldar, Apr 08 2023 (Start)
a(A005278(n)) = 0.
a(A131825(n)) = 1.
a(A063741(n)) = n. (End)

Name edited by Charles R Greathouse IV, Mar 17 2017

A362185 Numbers k with a single solution x to the equation A323410(x) = k.

Original entry on oeis.org

0, 4, 7, 9, 11, 216, 218, 220, 546, 652, 666, 700, 834, 850, 906, 924, 996, 1242, 1386, 1476, 1506, 1516, 1596, 1646, 1662, 1758, 1770, 1858, 1890, 1900, 1946, 2046, 2170, 2262, 2352, 2422, 2578, 2626, 2668, 2682, 2814, 2842, 2980, 2992, 3010, 3048, 3100, 3154

Offset: 1

Amiram Eldar, Apr 10 2023

nonn

Numbers k such that A362181(k) = 1.

			0 is a term since there is only one solution, x = 1, to A323410(x) = 0.

Amiram Eldar, Table of n, a(n) for n = 1..2335 (terms below 10^5)

Cf. A323410, A362181.
The unitary version of A131825.
Similar sequence: A361969.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; Join[{0}, Position[solnum, 1] // Flatten]]

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

Amiram Eldar, Apr 11 2023

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.

			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;

Amiram Eldar, Table of n, a(n) for n = 2..9674 (rows 2..1000)

Cf. A005278, A051953, A063507, A063740 (row lengths), A063741, A063742, A063748, A100827, A101373, A131825, A131826.

Similar sequences: A032447, A361966, A362180.

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]

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

Franz Vrabec, Jul 19 2007

nonn

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

Cf. A051953, A063742, A131825, A131826.

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

More terms from Jinyuan Wang, Mar 21 2020

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A063740 Number of integers k such that cototient(k) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A362185 Numbers k with a single solution x to the equation A323410(x) = k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Extensions