1,1

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.

Smallest prime solutions appear to be identical with A054906.

a(23) is presently unknown.

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

For n = 23, 35, 49, 59, 61, and 73, there are no terms up to 10^9. - Michel Marcus, Nov 02 2025

R. Sivaramakrishnan, Classical theory of Arithmetical Functions, (1989). Marcel Dekker, Inc., New York-Basel. Chapter V, Problem 20, page 113.

a(19) = 1058 because phi(1058 + 38) = phi(1096) = 544 = 506 + 38 = phi(1058) + 38.

a(100) = 225, phi(225 + 200) = phi(425) = 320 = 120 + 200 = phi(225) + 200.

A055458 := proc(n)

local x;

for x from 0 do

if not isprime(x) then

if numtheory[phi](x+2*n) = numtheory[phi](x)+2*n then

return x;

end if;

end if;

end do:

end proc: # R. J. Mathar, Sep 23 2016

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 *)

(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

Cf. A000010, A054906, A050472, A050473, A007694, A049237.

Sequence in context: A371139 A371145 A370989 * A381816 A360570 A178733

Adjacent sequences: A055455 A055456 A055457 * A055459 A055460 A055461

nonn,more

Labos Elemer, Jun 26 2000

More terms from Michel ten Voorde Jun 14 2003

Entry revised by N. J. A. Sloane, Apr 28 2015

approved