OFFSET
1,2
COMMENTS
Conjecture: a(n) is odd for all n. Verified up to n <= 3*10^5. - Jianing Song, Feb 21 2021
The conjecture above is false because a(16842752) = 33817088; see A002181 and A143510. - Flávio V. Fernandes, Oct 08 2023
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Ho-joo Lee and Gerald Myerson, Consecutive Integers Whose Totients Are Multiples of n: 10837, The American Mathematical Monthly, Vol. 110, No. 2 (Feb., 2003), pp. 158-159.
Pieter Moree and Hans Roskam, On an arithmetical function related to Euler's totient and the discriminator, Fib. Quart., Vol. 33, No. 4 (1995), pp. 332-340.
József Sándor, On the Euler minimum and maximum functions, Notes on Number Theory and Discrete Mathematics, Volume 15, 2009, Number 3, pp. 1—8.
FORMULA
Sequence is unbounded; a(n) <= n^2 since phi(n^2) is always divisible by n.
If n+1 is prime then a(n) = n+1.
a(n) = min{ k : phi(k) == 0 (mod n) }.
a(n) = a(2n) for odd n > 1. - Jianing Song, Feb 21 2021
EXAMPLE
a(48) = 65 because phi(65) = phi(5)*phi(13) = 4*12 = 48 and no smaller integer m has phi(m) divisible by 48.
MATHEMATICA
a = ConstantArray[1, 64]; k = 1; While[Length[vac = Rest[Flatten[Position[a, 1]]]] > 0, k++; a[[Intersection[Divisors[EulerPhi[k]], vac]]] *= k]; a (* Ivan Neretin, May 15 2015 *)
PROG
(PARI) a(n) = my(s=1); while(eulerphi(s)%n, s++); s;
vector(100, n, a(n))
(Python)
from sympy import totient as phi
def a(n):
k = 1
while phi(k)%n != 0: k += 1
return k
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Feb 21 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Melvin J. Knight (knightmj(AT)juno.com), May 25 2001
STATUS
approved