A176472 Smallest m for which A064380(m) - A000010(m) = n.

2, 4, 9, 12, 22, 18, 38, 16, 93, 45, 62, 70, 44, 63, 36, 52, 64, 102, 48, 68, 84, 76, 90, 142, 146, 117, 81, 166, 174, 178, 126, 80, 150, 132, 116, 230, 124, 100, 156, 246, 266, 258, 254, 148, 112

Offset: 0

Author

Vladimir Shevelev, Apr 18 2010

Keywords

• nonn

Comments

My 1981 publication studies A064380 with the quite natural convention A064380(1)=1. So a(1) could alternatively be defined as 1. By the definitions, it is clear that A064380(m) >= A000010(m).

Theorem. For every n >= 0, the equation A064380(m) - A000010(m) = n has infinitely many solutions.

References

• V. S. Abramovich (Shevelev), On an analog of the Euler function, Proceeding of the North-Caucasus Center of the Academy of Sciences of the USSR (Rostov na Donu), 2 (1981), 13-17.
• Vladimir Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..1000
• Simon Litsyn and Vladimir Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Crossrefs

Cf. A000010, A064380, A050376, A050292.

Programs

Maple

A176472 := proc(n) local m; for m from 2 do if A064380(m) - numtheory[phi](m) = n then return m; end if; end do: end proc: # R. J. Mathar, Jun 16 2010

Mathematica

infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[FactorInteger[g][[All, 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0&]]];
A064380[n_] := Sum[Boole[infCoprimeQ[j, n]], {j, 1, n - 1}];
a[n_] := a[n] = For[m = 2, True, m++, If[A064380[m] - EulerPhi[m] == n, Return[m]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 05 2023, after Amiram Eldar in A064380 *)

Extensions

a(2), a(3), a(8) and a(15) corrected and sequence extended by R. J. Mathar, Jun 16 2010