A307640 Least number k such that n divides gcd(sigma(k), phi(k), tau(k)).

1, 3, 18, 15, 3344, 45, 24128, 30, 882, 3344, 1012736, 126, 1953792, 24128, 16200, 168, 452263936, 2016, 1852571648, 3344, 40768, 1012736, 27007123456, 420, 1490000, 1953792, 103968, 24128, 2739920699392, 30096, 8348342681600, 840, 9114624, 452263936, 6163776, 2016

Offset: 1

Author

Marius A. Burtea, Apr 19 2019

Keywords

• nonn

Comments

For each n >= 1 there are infinitely many numbers s such that n divides sigma(s), phi(s) and tau(s).

From Dirichlet's theorem there are infinitely many numbers m for which the numbers p = n*m + 1 are prime. Then sigma(p^(n-1)), phi(p^(n-1)) and tau(p^(n-1)) numbers are divisible by n.

Examples

			For n = 2, sigma(3) = 4, phi(3) = 2, tau(3) = 4 are divisible by 2.
For n = 5, sigma(3344) = 7440, phi (3344) = 1440, tau (3344) = 20 are divisible by 5 and by 10.
For n = 11, sigma(1012736) = 2161632 = 11 * 196512, phi(1012736) = 11 * 43008, tau(1012736) = 11 * 4 are divisible by 11.
		

References

• Laurențiu Panaitopol, Alexandru Gica, Arithmetic problems and number theory. Ideas and methods of solving, Ed. Gil, Zalău, 2006, ch. 13, p. 79, pr. 18. (in Romanian).

Crossrefs

Cf. A000005, A000010, A000203, A009223, A222713.

Programs

Magma

for m in [1..16] do
      for n in [1..2000000] do
              if IsIntegral(SumOfDivisors(n)/m) and IsIntegral(EulerPhi(n)/m) and IsIntegral(NumberOfDivisors(n)/m) then
             m,n;
             break;
            end if;
      end for;
end for;

Mathematica

Array[Block[{i = 1}, While[Mod[GCD[DivisorSigma[1, i], EulerPhi@ i,DivisorSigma[0, i]], #] != 0, i++]; i] &, 16] (*Adaptation after A222713*)

PARI

isok(n,k) = ! frac(gcd(sigma(k), gcd(eulerphi(k), numdiv(k)))/n);
a(n) = my(k=1); while(!isok(n,k), k++); k; \\ Michel Marcus, Apr 20 2019

PARI

a(n) = {if(n==1,return(1)); my(res = oo, f = factor(n), hpf = f[#f~, 1]); forprime(p = 2, oo, if(p ^ (hpf - 1) > res, return(res)); forstep(i = p ^ (hpf - 1), res, p ^ (hpf - 1), if(isok(n, i), res = min(res, i);  next(2) ) ) ) } \\ uses isok from above \\ David A. Corneth, Apr 22 2019

Extensions

More terms from David A. Corneth, Apr 21 2019