A079277 Largest integer k < n such that any prime factor of k is also a prime factor of n.

1, 1, 2, 1, 4, 1, 4, 3, 8, 1, 9, 1, 8, 9, 8, 1, 16, 1, 16, 9, 16, 1, 18, 5, 16, 9, 16, 1, 27, 1, 16, 27, 32, 25, 32, 1, 32, 27, 32, 1, 36, 1, 32, 27, 32, 1, 36, 7, 40, 27, 32, 1, 48, 25, 49, 27, 32, 1, 54, 1, 32, 49, 32, 25, 64, 1, 64, 27, 64, 1, 64, 1, 64, 45, 64, 49, 72, 1, 64, 27

Offset: 2

Author

Istvan Beck (istbe(AT)online.no), Feb 07 2003

Keywords

• nonn
• look

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.

Penultimate term of row n in A162306. (The last term of row n in A162306 is n.)

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)

Examples

			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).
		

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.

Crossrefs

Cf. A000010, A007947, A051953, A162306, A208815, A272618, A285328, A285699, A285707, A285709, A285710, A285711.

Programs

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 *)

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

Formula

Largest k < n with rad(kn) = rad(n), where rad = A007947.