A055458 a(n) = smallest composite solution x to the equation phi(x+2n) = phi(x)+2n.

6, 12, 21, 24, 36, 45, 48, 39, 63, 72, 72, 95, 60, 57, 224, 84, 15, 135, 1058, 45, 301, 144

Offset

1,1

Comments

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

References

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

Links

Table of n, a(n) for n=1..22.

Example

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.

Maple

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

Mathematica

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A371139 A371145 A370989 * A360570 A178733 A344033

Adjacent sequences: A055455 A055456 A055457 * A055459 A055460 A055461

Keyword

nonn,more

Author

Labos Elemer, Jun 26 2000

Extensions

More terms from Michel ten Voorde Jun 14 2003

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

Status

approved