A196079 Difference between the largest and smallest inverse of totient function.

1, 3, 7, 11, 15, 11, 29, 43, 35, 41, 23, 55, 29, 31, 69, 89, 109, 55, 69, 47, 145, 53, 81, 87, 59, 137, 155, 67, 71, 197, 79, 207, 83, 165, 187, 141, 323, 149, 103, 159, 107, 269, 121, 235, 177, 319, 127, 255, 131, 253, 137, 139, 213, 445, 149, 151

Offset: 1

Author

Franz Vrabec, Sep 27 2011

Keywords

• nonn

Comments

No terms are zero if Carmichael's conjecture is true.

Even terms are rare: e.g., all inverses of 257*2^16 are even [Foster], so the difference between the largest and smallest inverse is even.

Examples

			Let n=3. The largest inverse of A002202(3)=4 is A006511(3)=12, the smallest inverse is A002181(3)=5, so a(3)=12-5=7.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• William P. Wardlaw, L. L. Foster and R. J. Simpson, Problem E3361, Amer. Math. Monthly, Vol. 98, No. 5 (May, 1991), 443-444.

Crossrefs

Cf. A002181, A002202, A006511.

Programs

Mathematica

max = 300; inversePhi[?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m] := Module[{p, nmax, n, nn}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; nn = Reap[While[n <= nmax, If[EulerPhi[n] == m, Sow[n]]; n++]] // Last; If[nn == {}, {}, First[nn]]]; Join[{2}, Reap[For[n = 2, n <= max, n = n + 2, nn = inversePhi[n] ; If[ nn != {} , Sow[Max[nn] - Min[nn]]]]] // Last // First] (* Jean-François Alcover, Nov 21 2013 *)

Formula

a(n) = A006511(n) - A002181(n).

Extensions

a(1) corrected by the editors, Nov 23 2013

a(1) in b-file corrected by Andrew Howroyd, Feb 22 2018