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A100827 Highly cototient numbers: records for a(n) in A063741. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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. [From T. D. Noe, Mar 16 2010]

LINKS

T. D. Noe, Table of n, a(n) for n=1..176

Wikipedia, Highly cototient number

EXAMPLE

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}.

MATHEMATICA

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]

CROSSREFS

Cf. A063741, A097942, A007374, A101373.

Sequence in context: A002075 A122623 A124014 * A034906 A018323 A151380

Adjacent sequences:  A100824 A100825 A100826 * A100828 A100829 A100830

KEYWORD

nonn

AUTHOR

Alonso del Arte, Jan 06 2005

EXTENSIONS

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

STATUS

approved

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Last modified April 24 14:49 EDT 2014. Contains 240985 sequences.