OFFSET
1,1
COMMENTS
Corresponding values of phi(k) are 1, 4, 8, 1296, 7776, 2985984, ...
Freiberg (Theorem 1.2) shows that there are >> (n*x^(1/n))/(log x)^(n+2) such values of k up to x. He calls the set of such numbers B*(x;-1;n). In particular, a(n) exists for each n.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..40
W. D. Banks and F. Luca, Power totients with almost primes, author's copy, Integers 11A (2011), 307-313.
Tristan Freiberg, Products of shifted primes simultaneously taking perfect power values, arXiv:1008.1978 [math.NT], 2010.
Tristan Freiberg and Carl Pomerance, A note on square totients, International Journal of Number Theory, Volume 11, Issue 08, December 2015.
EXAMPLE
a(4) = 3458 = 2 * 7 * 13 * 19 and phi(3458) = (2-1)*(7-1)*(13-1)*(19-1) = 6^4.
PROG
(PARI) a(n) = {my(k=2); while (!(issquarefree(k) && (omega(k)==n) && (ispower(eulerphi(k), n))), k++); k; } \\ Michel Marcus, Jan 16 2017
(Python)
from itertools import count
from math import isqrt, prod
from sympy import perfect_power, primefactors, primerange, integer_nthroot, primepi
def A281069(n):
def squarefreealmostprimepi(n, k):
if k==0: return int(n>=1)
def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b+1, isqrt(x//c)+1), a+1)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b+1, integer_nthroot(x//c, m)[0]+1), a+1) for d in g(x, a2, b2, c*b2, m-1)))
return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n, 0, 1, 1, k)) if k>1 else primepi(n))
return next(k for k in (squarefreealmostprime(i, n) for i in count(1)) if (p:=perfect_power(prod(p-1 for p in primefactors(k)))) and p[1] == n) if n>1 else 2 # Chai Wah Wu, Sep 09 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Altug Alkan, Jan 14 2017
EXTENSIONS
a(9)-a(10) from Jinyuan Wang, Nov 08 2020
a(11)-a(13) from Daniel Suteu confirmed and a(14)-a(15) added by Max Alekseyev, Jul 07 2026
More terms from David A. Corneth, Jul 08 2026
STATUS
approved