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%I #39 Jan 08 2019 09:12:41
%S 6,12,21,24,36,45,48,39,63,72,72,95,60,57,224,84,15,135,1058,45,301,
%T 144
%N a(n) = smallest composite solution x to the equation phi(x+2n) = phi(x)+2n.
%C Sivaramakrishnan (1989) quotes Makowski, who gave solutions for phi(x+d) = phi(x)+d with d = 2^a and d = 2*3^a. Compare also A007694 and A049237.
%C Smallest prime solutions appear to be identical with A054906.
%C a(23) is presently unknown.
%C The sequence continues as (with ? for unknown values): ?, 95, 162, 63, 189, 69, 156, 161, 180, 69, 260, 150, ?, 115, 204, 129, 400, 75, 180, 165, 35, 117, 476, 7105, 288, 195, ?, 324, 620, 240, 81, 145, 14531, 153, 644, 309, ?, 203, ?, 63, 640, 75, 372, 285, 2312, 33, 343, 642, 336, 155, ?, 147, 728, 396, 1564, 185, 564, 87, 567, 360, 360, 155, 492, 510, 560, 516, 516, 301, 4232, 261, 860, 387, 576, 185, 564, 309, 1000, 225 ... - _Don Reble_, Apr 29 2015
%D Sivaramakrishnan, R. (1989): Classical theory of Arithmetical Functions. Marcel Dekker, Inc., New York-Basel. Chapter V, Problem 20, page 113.
%e a(19) = 1058 because phi(1058 + 38) = phi(1096) = 544 = 506 + 38 = phi(1058) + 38.
%e a(100) = 225, phi(225 + 200) = phi(425) = 320 = 120 + 200 = phi(225) + 200.
%p A055458 := proc(n)
%p local x;
%p for x from 0 do
%p if not isprime(x) then
%p if numtheory[phi](x+2*n) = numtheory[phi](x)+2*n then
%p return x;
%p end if;
%p end if;
%p end do:
%p end proc: # _R. J. Mathar_, Sep 23 2016
%t Table[k = 4; While[Nand[CompositeQ@ k, EulerPhi[k + 2 n] == EulerPhi[k] + 2 n], k++]; k, {n, 22}] (* _Michael De Vlieger_, Dec 17 2016 *)
%o (PARI) a(n)=forcomposite(x=4,, if(eulerphi(x+2*n) == eulerphi(x)+2*n, return(x))) \\ does not handle -1s; _Charles R Greathouse IV_, Apr 28 2015
%Y Cf. A000010, A054906, A050472, A050473, A007694, A049237.
%K nonn,more
%O 1,1
%A _Labos Elemer_, Jun 26 2000
%E More terms from _Michel ten Voorde_ Jun 14 2003
%E Entry revised by _N. J. A. Sloane_, Apr 28 2015
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