A079277 - OEIS

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

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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.

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)

LINKS

David W. Wilson, Table of n, a(n) for n = 2..10000

Aled Walker and Alexander Walker, Arithmetic Progressions with Restricted Digits, Amer. Math. Monthly 127(2) (2020). See also arXiv:1809.02430 [math.NT], 2018. See Prop. 2.

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