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

%I A222713 #45 Dec 15 2024 19:42:01
%S A222713 1,3,14,12,88,14,116,15,190,88,989,35,477,116,209,105,6901,190,7067,
%T A222713 88,196,989,6439,35,15049,477,2754,172,10207,209,4976,336,989,6901,
%U A222713 1189,190,10877,7067,477,248,13529,377,44461,989,418,6439,79523,105,10244,15049
%N A222713 Least number k such that n divides gcd(sigma(k), phi(k)) (A009223).
%C A222713 For each n there are infinitely many numbers k for which n divides sigma(k) and phi(k). - _Marius A. Burtea_, Mar 28 2019
%H A222713 Giovanni Resta, <a href="/A222713/b222713.txt">Table of n, a(n) for n = 1..10000</a> (terms 1-388 from  Marius A. Burtea, 389-2808 from David A. Corneth)
%e A222713 Given A009223 = 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 4, 2, 6, 8, 1, 2, 3, ...
%e A222713 1 first divides A009223(1); 2 first divides A009223(3); 3 first divides A009223(14)=6.
%t A222713 Array[Block[{i = 1}, While[Mod[GCD[DivisorSigma[1, i], EulerPhi@ i], #] != 0, i++]; i] &, 50] (* _Michael De Vlieger_, Mar 28 2019 *)
%o A222713 (PARI) a(n)={my(k=1); while(gcd(sigma(k), eulerphi(k))%n!=0, k++); k}
%o A222713 (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
%o A222713 (Magma) v:=[];
%o A222713 for n in [1..60] do
%o A222713 m:=1;
%o A222713         while  not EulerPhi(m) mod n  eq 0 or not SumOfDivisors(m) mod n  eq 0 do
%o A222713            v[n]:=0;
%o A222713            m:=m+1;
%o A222713         end while;
%o A222713      v[n]:=m;
%o A222713 end for;
%o A222713 v; // _Marius A. Burtea_, Mar 30 2019
%Y A222713 Cf. A009223, A222714.
%K A222713 nonn
%O A222713 1,2
%A A222713 _Phil Carmody_, Mar 01 2013