A063748 -id:A063748 - cp's OEIS frontend

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

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

A063507 Least k such that k - phi(k) = n, or 0 if no such k exists.

Original entry on oeis.org

2, 4, 9, 6, 25, 10, 15, 12, 21, 0, 35, 18, 33, 26, 39, 24, 65, 34, 51, 38, 45, 30, 95, 36, 69, 0, 63, 52, 161, 42, 87, 48, 93, 0, 75, 54, 217, 74, 99, 76, 185, 82, 123, 60, 117, 66, 215, 72, 141, 0, 235, 0, 329, 78, 159, 98, 105, 0, 371, 84, 177, 122, 135, 96, 305, 90, 427

Offset: 1

Labos Elemer, Aug 09 2001

nonn

Inverse cototient (A051953) sets represented by their minimum, as in A002181 for totient function. Impossible values (A005278) are replaced by zero.

If a(n) > 0, then it appears that a(n) > 1.26n. - T. D. Noe, Dec 06 2006

			x = InvCototient[24] = {36, 40, 44, 46}; Phi[x] = Phi[{36, 40, 44, 46}] = {12, 16, 20, 22}; x-Phi[x] = {24, 24, 24, 24}, so a(24) = Min[InvCototient[24]]; a(10) = 0 because 10 is in A005278.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A051953, A000010, A002181, A005277, A005278.
Cf. A063748 (greatest solution to x-phi(x)=n).
Cf. A063740 (number of k such that cototient(k) = n).

Table[SelectFirst[Range[n^2 + 1], # - EulerPhi[#] == n &] /. k_ /; ! IntegerQ@ k -> 0, {n, 67}] (* Michael De Vlieger, Jan 11 2018 *)

a(n)-A051953(a(n)) = n if possible and a(n)=0 if n belongs to A005278.

Edited by N. J. A. Sloane, Oct 25 2008 at the suggestion of R. J. Mathar

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]

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

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A063507 Least k such that k - phi(k) = n, or 0 if no such k exists.

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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.

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