A208815 -id:A208815 - cp's OEIS frontend

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

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

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)

			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).

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.

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

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 *)

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

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

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

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

A285710 Numbers n for which A000010(n) = A285699(n); positions of zeros in A285709.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 16, 17, 19, 21, 23, 25, 27, 28, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 62, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 124, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 237

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

After a(1) = 1, also numbers n such that A051953(n) = A079277(n).

Antti Karttunen, Table of n, a(n) for n = 1..6672

Positions of zeros in A285709.
Cf. A000010, A051953, A079277, A208815, A285699.
Cf. A000961 (a subsequence).

Flatten@ Position[#, 0] &@ Table[EulerPhi@ n - (n - If[n <= 2, n - 1, Module[{k = n - 2, e = Floor@ Log2@ n}, While[PowerMod[n, e, k] != 0, k--]; k]]), {n, 240}] (* Michael De Vlieger, Apr 26 2017 *)

from sympy import divisors, totient
from sympy.ntheory.factor_ import core
def a007947(n): return max(i for i in divisors(n) if core(i) == i)
def a079277(n):
    k=n - 1
    while True:
        if a007947(k*n) == a007947(n): return k
        else: k-=1
def a285699(n): return 1 if n<2 else n - a079277(n)
print([n for n in range(1, 301) if totient(n) == a285699(n)]) # Indranil Ghosh, Apr 26 2017

A285709 a(n) = A000010(n) - A285699(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 4, 0, 4, 0, 4, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 14, 14, 14, 8, 0, 12, 12, 8, 0, 6, 0, 8, 6, 8, 0, 4, 0, 10, 8, 4, 0, 12, 10, 17, 6, 2, 0, 10, 0, 0, 22, 0, 8, 18, 0, 28, 2, 18, 0, 16, 0, 26, 10, 24, 32, 18, 0, 16, 0, 22, 0, 21, 4, 20, 50, 16, 0, 15, 30, 16, 48, 16, 2, 17, 0, 8, 42, 20, 0, 26, 0, 8, 24, 10, 0, 24, 0, 30, 42, 34, 0, 30, -2

Offset: 1

Antti Karttunen, Apr 26 2017

The scatter plot has unusual "rays".

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A007947, A051953, A079277, A285699, A285711.

Cf. A285710 (positions of zeros), A208815 (of negative terms).

Table[EulerPhi@ n - (n - If[n <= 2, n - 1, Module[{k = n - 2, e = Floor@ Log2@ n}, While[PowerMod[n, e, k] != 0, k--]; k]]), {n, 115}] (* Michael De Vlieger, Apr 26 2017 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ From 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)); };
A285709(n) = if(!n,n,(eulerphi(n)+A079277(n))-n);

from sympy import divisors, totient
from sympy.ntheory.factor_ import core
def a007947(n): return max(i for i in divisors(n) if core(i) == i)
def a079277(n):
    k=n - 1
    while True:
        if a007947(k*n) == a007947(n): return k
        else: k-=1
def a285699(n): return 1 if n<2 else n - a079277(n)
def a(n): return totient(n) - a285699(n)
print([a(n) for n in range(1, 116)]) # Indranil Ghosh, Apr 26 2017

(define (A285709 n) (- (A000010 n) (A285699 n)))

a(n) = A000010(n) - A285699(n).

For n > 1, a(n) = (A000010(n) + A079277(n)) - n = A079277(n) - A051953(n).

cp's OEIS Frontend

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

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A285710 Numbers n for which A000010(n) = A285699(n); positions of zeros in A285709.

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A285709 a(n) = A000010(n) - A285699(n).

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