A285711 -id:A285711 - cp's OEIS frontend

A009195 a(n) = gcd(n, phi(n)).

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 8, 5, 2, 9, 4, 1, 2, 1, 16, 1, 2, 1, 12, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 16, 7, 10, 1, 4, 1, 18, 5, 8, 3, 2, 1, 4, 1, 2, 9, 32, 1, 2, 1, 4, 1, 2, 1, 24, 1, 2, 5, 4, 1, 6, 1, 16, 27, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 32, 1, 14, 3, 20

Offset: 1

David W. Wilson

The inequality gcd(n, phi(n)) <= 2n exp(-sqrt(log 2 log n)) holds for all squarefree n >= 1 (Erdős, Luca, and Pomerance).

Erdős shows that for almost all n, a(n) ~ log log log log n. - Charles R Greathouse IV, Nov 23 2011

T. D. Noe, Table of n, a(n) for n = 1..10000
Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.) 12 (1948), pp. 75-78.
Paul Erdős, Florian Luca, Carl Pomerance, On the proportion of numbers coprime to a given integer, in Anatomy of Integers, pp. 47-64, J.-M. De Koninck, A. Granville, F. Luca (editors), AMS, 2008.
Joshua Stucky, The distribution of gcd(n,phi(n)), arXiv:2402.13997 [math.NT], 2024.

Cf. A000010, A003277, A050399, A051953, A061303, A109395, A285711.

a009195 n = n `gcd` a000010 n  -- Reinhard Zumkeller, Feb 27 2012

[Gcd(n, EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Dec 17 2015

a009195 := n -> igcd(i,numtheory[phi](n));

Table[GCD[n,EulerPhi[n]],{n,100}] (* Harvey P. Dale, Aug 11 2011 *)

a(n)=gcd(n,eulerphi(n)) \\ Charles R Greathouse IV, Nov 23 2011

def a009195(n):
    from math import gcd
    phi = lambda x: len([i for i in range(x) if gcd(x,i) == 1])
    return gcd(n, phi(n))
# Edward Minnix III, Dec 05 2015

a(n) = gcd(n, A051953(n)). - Labos Elemer

a(n) = n / A109395(n). - Antti Karttunen, May 04 2017 (corrected also typo in above formula).

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

Original entry on oeis.org

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.

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).

A285707 a(n) = gcd(n, A079277(n)), a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 3, 1, 2, 3, 8, 1, 2, 1, 4, 3, 2, 1, 6, 5, 2, 9, 4, 1, 3, 1, 16, 3, 2, 5, 4, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 12, 7, 10, 3, 4, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 32, 5, 2, 1, 4, 3, 2, 1, 8, 1, 2, 15, 4, 7, 6, 1, 16, 27, 2, 1, 3, 5, 2, 3, 8, 1, 9, 7, 4, 3, 2, 5, 3, 1, 2, 9, 20, 1, 6, 1, 8, 3, 2, 1, 12, 1, 10, 3, 14, 1, 6, 5, 4, 9

Offset: 1

Antti Karttunen, Apr 26 2017

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

Cf. A009195, A079277, A285699, A285708, A285711.

Table[GCD[n, #] &@ If[n <= 2, 1, Module[{k = n - 2, e = Floor@ Log2@ n}, While[PowerMod[n, e, k] != 0, k--]; k]], {n, 117}] (* Michael De Vlieger, Apr 26 2017 *)

A007947(n) = factorback(factorint(n)[, 1]);
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)); };
A285707(n) = if(1==n,n,gcd(A079277(n),n));

from sympy import divisors, gcd
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 a(n): return 1 if n==1 else gcd(n, a079277(n))
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Apr 26 2017

(define (A285707 n) (if (= 1 n) n (gcd n (A079277 n))))

a(1) = 1; for n > 1, a(n) = gcd(n, A079277(n)) = gcd(n, A285699(n)).

a(n) = n / A285708(n).

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A009195 a(n) = gcd(n, phi(n)).

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

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

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A285707 a(n) = gcd(n, A079277(n)), a(1) = 1.

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