A222713 Least number k such that n divides gcd(sigma(k), phi(k)) (A009223).

1, 3, 14, 12, 88, 14, 116, 15, 190, 88, 989, 35, 477, 116, 209, 105, 6901, 190, 7067, 88, 196, 989, 6439, 35, 15049, 477, 2754, 172, 10207, 209, 4976, 336, 989, 6901, 1189, 190, 10877, 7067, 477, 248, 13529, 377, 44461, 989, 418, 6439, 79523, 105, 10244, 15049

Offset: 1

Author

Phil Carmody, Mar 01 2013

Keywords

• nonn

Comments

For each n there are infinitely many numbers k for which n divides sigma(k) and phi(k). - Marius A. Burtea, Mar 28 2019

Examples

			Given A009223 = 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 4, 2, 6, 8, 1, 2, 3, ...
1 first divides A009223(1); 2 first divides A009223(3); 3 first divides A009223(14)=6.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (terms 1-388 from Marius A. Burtea, 389-2808 from David A. Corneth)

Crossrefs

Cf. A009223, A222714.

Programs

Magma

[Min([n: n in [1..300000] | IsIntegral(SumOfDivisors(n)/m) and IsIntegral(EulerPhi(n)/m) ]): m in [1..70]]; // Marius A. Burtea, Mar 28 2019

Magma

v:=[];
for n in [1..60] do
m:=1;
        while  not EulerPhi(m) mod n  eq 0 or not SumOfDivisors(m) mod n  eq 0 do
           v[n]:=0;
           m:=m+1;
        end while;
     v[n]:=m;
end for;
v; // Marius A. Burtea, Mar 30 2019

Mathematica

Array[Block[{i = 1}, While[Mod[GCD[DivisorSigma[1, i], EulerPhi@ i], #] != 0, i++]; i] &, 50] (* Michael De Vlieger, Mar 28 2019 *)

PARI

a(n)={my(k=1); while(gcd(sigma(k), eulerphi(k))%n!=0, k++); k}