OFFSET
2,3
COMMENTS
The function a(n) complements Euler's phi-function: 1) a(n)+phi(n) = n if n is a power of a prime (actually, in A285710). 2) It seems also that a(n)+phi(n) >= n for "almost all numbers" (see A285709, A208815). 3) a(2n) = n+1 if and only if n is a Mersenne prime. 4) Lim a(n^k)/n^k =1 if n has at least two prime factors and k goes to infinity.
From Michael De Vlieger, Apr 26 2017: (Start)
In other words, largest integer k < n such that k | n^e with integer e >= 0.
For prime p, a(p) = 1. More generally, for n with omega(n) = 1, that is, a prime power p^e with e > 0, a(p^e) = p^(e - 1).
For n with omega(n) > 1, a(n) does not divide n. If n = pq with q = p + 2, then p^2 < n though p^2 does not divide n, yet p^2 | n^e with e > 1. If n has more than 2 distinct prime divisors p, powers p^m of these divisors will appear in the range (1..n-1) such that p^m > n/lpf(n) (lpf(n) = A020639(n)). Since a(n) is the largest of these, a(n) is not a divisor of n.
If a(n) does not divide n, then a(n) appears last in row n of A272618.
(End)
LINKS
David W. Wilson, Table of n, a(n) for n = 2..10000
Aled Walker and Alexander Walker, Arithmetic Progressions with Restricted Digits, arXiv:1809.02430 [math.NT], 2018.
FORMULA
Largest k < n with rad(kn) = rad(n), where rad = A007947.
EXAMPLE
a(10)=8 since 8 is the largest integer< 10 that can be written using only the primes 2 and 5. a(78)=72 since 72 is the largest number less than 78 that can be written using only the primes 2, 3 and 13. (78=2*3*13).
MATHEMATICA
Table[If[n == 2, 1, Module[{k = n - 2, e = Floor@ Log2@ n}, While[PowerMod[n, e, k] != 0, k--]; k]], {n, 2, 81}] (* Michael De Vlieger, Apr 26 2017 *)
PROG
(PARI) a(n) = {forstep(k = n - 1, 2, -1, f = factor(k); okk = 1; for (i=1, #f~, if ((n % f[i, 1]) != 0, okk = 0; break; )); if (okk, return (k)); ); return (1); } \\ Michel Marcus, Jun 11 2013
(PARI)
A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
A079277(n) = { my(r); if((n > 1 && !bitand(n, (n-1))), (n/2), r=A007947(n); if(1==n, 0, k = n-1; while(A007947(k*n) <> r, k = k-1); k)); }; \\ Antti Karttunen, Apr 26 2017
(Python)
from sympy import divisors
from sympy.ntheory.factor_ import core
def a007947(n): return max(d for d in divisors(n) if core(d) == d)
def a(n):
k=n - 1
while True:
if a007947(k*n) == a007947(n): return k
else: k-=1
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Apr 26 2017
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Istvan Beck (istbe(AT)online.no), Feb 07 2003
STATUS
approved