A063741 -id:A063741 - cp's OEIS frontend

A100827 Highly cototient numbers: records for a(n) in A063741.

2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889, 2099, 2309, 2729, 3149, 3359, 3569, 3989, 4199, 4289, 4409, 4619, 5249, 5459, 5879, 6089, 6509, 6719, 6929

Offset: 1

Alonso del Arte, Jan 06 2005

nonn

Each number k on this list has more solutions to the equation x - phi(x) = k (where phi is Euler's totient function, A000010) than any preceding k except 1.
This sequence is a subset of A063741. As noted in that sequence, there are infinitely many solutions to x - phi(x) = 1. Unlike A097942, the highly totient numbers, this sequence has many odd numbers besides 1.
With the expection of 2, 4, 8, all of the known terms are congruent to -1 mod a primorial (A002110). The specific primorial satisfying this congruence would result in a sequence similar to A080404 a(n)=A007947[A055932(n)]. - Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Dec 28 2006
Because most of the solutions to x - phi(x) = k are semiprimes p*q with p+q=k+1, it appears that this sequence eventually has terms that are one less than the Goldbach-related sequence A082917. In fact, terms a(108) to a(176) are A082917(n)-1 for n=106..174. [T. D. Noe, Mar 16 2010] This holds through a(229).  [Jud McCranie, May 18 2017]

			a(3) = 8 since x - phi(x) = 8 has three solutions, {12, 14, 16}, one more than a(2) = 4 which has two solutions, {6, 8}.

Jud McCranie, Table of n, a(n) for n = 1..229 (terms 1..176 by T. D. Noe)
Wikipedia, Highly cototient number

Cf. A063741, A097942, A007374, A101373.

searchMax = 4000; coPhiAnsYldList = Table[0, {searchMax}]; Do[coPhiAns = m - EulerPhi[m]; If[coPhiAns <= searchMax, coPhiAnsYldList[[coPhiAns]]++ ], {m, 1, searchMax^2}]; highlyCototientList = {2}; currHigh = 2; Do[If[coPhiAnsYldList[[n]] > coPhiAnsYldList[[currHigh]], highlyCototientList = {highlyCototientList, n}; currHigh = n], {n, 2, searchMax}]; Flatten[highlyCototientList]

More terms from Robert G. Wilson v, Jan 08 2005

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

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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A100827 Highly cototient numbers: records for a(n) in A063741.

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

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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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