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A307640 Least number k such that n divides gcd(sigma(k), phi(k), tau(k)). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For each n >= 1 there is an infinite numbers of s for which n divide sigma(s), phi(s) and tau(s).

From Dirichlet's theorem there is an infinity of 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.

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).

LINKS

Table of n, a(n) for n=1..36.

EXAMPLE

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.

MATHEMATICA

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

PROG

(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;

(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

CROSSREFS

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

Sequence in context: A077104 A073815 A212994 * A195998 A291167 A174029

Adjacent sequences:  A307637 A307638 A307639 * A307641 A307642 A307643

KEYWORD

nonn

AUTHOR

Marius A. Burtea, Apr 19 2019

EXTENSIONS

More terms from David A. Corneth, Apr 21 2019

STATUS

approved

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Last modified September 29 21:59 EDT 2020. Contains 337432 sequences. (Running on oeis4.)