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A306146 Numbers k such that A000010(A023900(k)) = A023900(A000010(k)).

%I A306146 #59 Feb 19 2020 07:23:59
%S A306146 1,14,22,28,44,46,56,75,88,92,94,112,118,166,176,184,188,214,224,236,
%T A306146 332,334,352,358,368,375,376,422,428,448,454,472,526,639,662,664,668,
%U A306146 694,704,716,718,736,752,766,844,856,867,896,908,926,934,944,958,1006,1052,1075,1094,1126,1142,1174,1179,1324
%N A306146 Numbers k such that A000010(A023900(k)) = A023900(A000010(k)).
%C A306146 No term is a product of an odd number of distinct prime factors (because then A023900 is negative, i.e., contains no terms from A030230).
%C A306146 For known terms:
%C A306146 - a(n) is nonsquarefree iff A000010(n) is nonsquarefree.
%C A306146 - If a(n) is squarefree then A000010(n) and A023900(n) are both squarefree.
%H A306146 Amiram Eldar, <a href="/A306146/b306146.txt">Table of n, a(n) for n = 1..10000</a>
%e A306146 75 is a term because A000010(A023900(75)) = A023900(A000010(75)) = 4.
%p A306146 isA306146 := proc(n)
%p A306146     local a239 ;
%p A306146     a239 := A023900(n) ;
%p A306146     if a239 >= 1 then
%p A306146         simplify( numtheory[phi](a239) = A023900(numtheory[phi](n)) );
%p A306146     else
%p A306146         false;
%p A306146     end if;
%p A306146 end proc:
%p A306146 for n from 1 to 1000 do
%p A306146     if isA306146(n) then
%p A306146         printf("%d,",n) ;
%p A306146     end if;
%p A306146 end do: # _R. J. Mathar_, Feb 14 2019
%t A306146 f[p_, e_] := 1 - p; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[1324],(d1 = d[#]) > 0 && d[EulerPhi[#]] == EulerPhi[d1] &] (* _Amiram Eldar_, Feb 19 2020 *)
%o A306146 (PARI) a023900(n) = sumdivmult(n, d, d*moebius(d))
%o A306146 is(n) = sdm = a023900(n); if(sdm < 0, return(0), sdmphi = a023900(eulerphi(n)); eulerphi(sdm) == sdmphi) \\ _David A. Corneth_, Aug 17 2018
%Y A306146 Cf. A000010, A023900, A030230.
%K A306146 nonn
%O A306146 1,2
%A A306146 _Torlach Rush_, Aug 11 2018