A316785 Numerator of an upper bound for the maximal element in phi^(-1)(n).

2, 6, 6, 15, 10, 21, 14, 30, 18, 33, 22, 455, 26, 42, 30, 255, 34, 133, 38, 165, 42, 69, 46, 455, 50, 78, 54, 435, 58, 2387, 62, 255, 66, 102, 70, 319865, 74, 114, 78, 1353, 82, 301, 86, 345, 90, 141, 94, 7735, 98, 165, 102, 795, 106, 399, 110, 435, 114, 177, 118, 1892891

Offset: 1

Author

Franz Vrabec, Jul 13 2018

Keywords

• nonn
• frac

Comments

A057635(n) <= a(n)/A316786(n).

Examples

			For n = 12, there are 5 primes p with (p-1)|12: p1 = 2, p2 = 3, p3 = 5, p4 = 7, and p5 = 13. The numerator of 12*(2/1)*(3/2)*(5/4)*(7/6)*(13/12) = 455/8 is a(12) = 455.
		

Links

• R. Coleman, Some remarks on Euler's totient function, HAL Id: hal-00718975, version 1, 2012.
• H. Gupta, Euler's totient function and its inverse, Indian J. Pure Appl. Math., 12(1) (1981), 22-29.

Crossrefs

Cf. A057635, A316786 (denominators).

Programs

Maple

with(numtheory): A316785 := proc(n) local d,N; N:=n; for d in divisors(n) do if is prime(d+1) then N := (N*(d+1))/(d) end if; end do; numer(N); end proc;

Mathematica

a[n_] := Block[{p = Select[Prime@ Range@ PrimePi[n + 1], Mod[n, # - 1] == 0 &]}, Numerator[n*Times @@ (p/(p - 1))]]; Array[a, 60] (* Robert G. Wilson v, Aug 01 2018 *)

PARI

a(n) = my(p=n); fordiv(n, d, if (isprime(d+1), p *= (d+1)/d)); numerator(p); \\ Michel Marcus, Jul 29 2018

Formula

Numerator of n*Product_{p prime, (p-1)|n} p/(p-1).