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

%I A176472 #26 Sep 06 2023 01:27:35
%S A176472 2,4,9,12,22,18,38,16,93,45,62,70,44,63,36,52,64,102,48,68,84,76,90,
%T A176472 142,146,117,81,166,174,178,126,80,150,132,116,230,124,100,156,246,
%U A176472 266,258,254,148,112
%N A176472 Smallest m for which A064380(m) - A000010(m) = n.
%C A176472 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).
%C A176472 Theorem. For every n >= 0, the equation A064380(m) - A000010(m) = n has infinitely many solutions.
%D A176472 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.
%D A176472 Vladimir Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.
%H A176472 Amiram Eldar, <a href="/A176472/b176472.txt">Table of n, a(n) for n = 0..1000</a>
%H A176472 Simon Litsyn and Vladimir Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponent</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.
%p A176472 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
%t A176472 infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[FactorInteger[g][[All, 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0&]]];
%t A176472 A064380[n_] := Sum[Boole[infCoprimeQ[j, n]], {j, 1, n - 1}];
%t A176472 a[n_] := a[n] = For[m = 2, True, m++, If[A064380[m] - EulerPhi[m] == n, Return[m]]];
%t A176472 Table[Print[n, " ", a[n]]; a[n], {n, 0, 100}] (* _Jean-François Alcover_, Sep 05 2023, after _Amiram Eldar_ in A064380 *)
%Y A176472 Cf. A000010, A064380, A050376, A050292.
%K A176472 nonn
%O A176472 0,1
%A A176472 _Vladimir Shevelev_, Apr 18 2010
%E A176472 a(2), a(3), a(8) and a(15) corrected and sequence extended by _R. J. Mathar_, Jun 16 2010