A316786 Denominator of an upper bound for the maximal element in phi^(-1)(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 20, 1, 2, 1, 1, 1, 1728, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 32, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 5760, 1, 1, 1, 1, 1, 44, 1, 1, 1, 10, 1, 62208, 1, 1, 1, 1, 1, 2, 1, 64, 1, 1, 1, 192, 1, 1, 1, 88, 1, 120, 1, 2, 1, 1, 1, 1536, 1, 1, 1, 8

Offset: 1

Author

Franz Vrabec, Jul 13 2018

Keywords

• nonn
• frac

Comments

A057635(n) <= A316785(n)/a(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 denominator of 12*(2/1)*(3/2)*(5/4)*(7/6)*(13/12) = 455/8 is a(12) = 8.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..5039
• 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.
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Crossrefs

Cf. A057635, A316785 (numerators).

Programs

Maple

with(numtheory): A316786 := 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; denom(N); end proc;

PARI

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

Formula

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