A097942 - OEIS

A097942

Highly totient numbers: each number k on this list has more solutions to the equation phi(x) = k than any preceding k (where phi is Euler's totient function, A000010).

1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 25920, 30240, 34560, 40320, 51840, 60480, 69120, 80640, 103680, 120960, 161280, 181440, 207360, 241920, 362880, 483840, 725760, 967680 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`If you inspect PhiAnsYldList after running the Mathematica program below, the zeros with even-numbered indices should correspond to the nontotients (A005277).` `Where records occur in A014197. - T. D. Noe, Jun 13 2006` `Cf. A131934.`
	LINKS	`Jud McCranie, Table of n, a(n) for n = 1..109 (terms 1..79 from T. D. Noe, terms 80..86 from Donovan Johnson)` `Wikipedia, Highly totient number.`
	EXAMPLE	`a(4) = 8 since phi(x) = 8 has the solutions {15, 16, 20, 24, 30}, one more solution than a(3) = 4 for which phi(x) = 4 has solutions {5, 8, 10, 12}.`
	MAPLE	`HighlyTotientNumbers := proc(n) # n > 1 is search maximum` `local L, m, i, r; L := NULL; m := 0;` `for i from 1 to n do` `r := nops(numtheory[invphi](i));` `if r > m then L := L, [i, r]; m := r fi` `od; [L] end:` `A097942_list := n -> seq(s[1], s = HighlyTotientNumbers(n));` `A097942_list(500); # Peter Luschny, Sep 01 2012`
	MATHEMATICA	`searchMax = 2000; phiAnsYldList = Table[0, {searchMax}]; Do[phiAns = EulerPhi[m]; If[phiAns <= searchMax, phiAnsYldList[[phiAns]]++ ], {m, 1, searchMax^2}]; highlyTotientList = {1}; currHigh = 1; Do[If[phiAnsYldList[[n]] > phiAnsYldList[[currHigh]], highlyTotientList = {highlyTotientList, n}; currHigh = n], {n, 2, searchMax}]; Flatten[highlyTotientList]`
	PROG	`(Sage)` `def HighlyTotientNumbers(n) : # n > 1 is search maximum.` `R = {}` `for i in (1..n^2) :` `r = euler_phi(i)` `if r <= n :` `R[r] = R[r] + 1 if r in R else 1` `# print R.keys() # A002202` `# print R.values() # A058277` `P = []; m = 1` `for l in sorted(R.keys()) :` `if R[l] > m : m = R[l]; P.append((l, m))` `# print [l[0] for l in P] # A097942` `# print [l[1] for l in P] # A131934` `return P` `A097942_list = lambda n: [s[0] for s in HighlyTotientNumbers(n)]` `A097942_list(500) # Peter Luschny, Sep 01 2012` `(PARI)` `{ A097942_list(n) = local(L, m, i, r);` `m = 0;` `for(i=1, n,` `\\ from Max Alekseyev, http://home.gwu.edu/~maxal/gpscripts/` `r = numinvphi(i);` `if(r > m, print1(i, ", "); m = r) );` `} \\ Peter Luschny, Sep 01 2012`
	CROSSREFS	`A subsequence of A007374.` `Cf. A000010, A005277, A014573, A004653, A105207, A105208.` `Sequence in context: A326076 A181808 A343014 * A354541 A358513 A004653` `Adjacent sequences: A097939 A097940 A097941 * A097943 A097944 A097945`
	KEYWORD	`nonn`
	AUTHOR	`Alonso del Arte, Sep 05 2004`
	EXTENSIONS	`Edited and extended by Robert G. Wilson v, Sep 07 2004`
	STATUS	`approved`