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%I #27 May 18 2017 20:10:09
%S 2,4,8,23,35,47,59,63,83,89,113,119,167,209,269,299,329,389,419,509,
%T 629,659,779,839,1049,1169,1259,1469,1649,1679,1889,2099,2309,2729,
%U 3149,3359,3569,3989,4199,4289,4409,4619,5249,5459,5879,6089,6509,6719,6929
%N Highly cototient numbers: records for a(n) in A063741.
%C 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.
%C 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.
%C 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
%C 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]
%H Jud McCranie, <a href="/A100827/b100827.txt">Table of n, a(n) for n = 1..229</a> (terms 1..176 by T. D. Noe)
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Highly_cototient_number">Highly cototient number</a>
%e 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}.
%t 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]
%Y Cf. A063741, A097942, A007374, A101373.
%K nonn
%O 1,1
%A _Alonso del Arte_, Jan 06 2005
%E More terms from _Robert G. Wilson v_, Jan 08 2005
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