A286591 -id:A286591 - cp's OEIS frontend

A009194 a(n) = gcd(n, sigma(n)).

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 28, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2

Offset: 1

David W. Wilson

nonn

LCM of common divisors of n and sigma(n). It equals n if n is multiply perfect (A007691). - Labos Elemer, Aug 14 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Pollack, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J. Volume 60, Issue 1 (2011), 199-214.

Cf. A000203, A003624, A007691, A014567, A017666, A063906, A069059, A073802, A082062, A179931, A205523, A216793 (positions of records), A234367, A249917.

Cf. also A009191, A009205, A009242, A274382, A286591, A286594.

a009194 n = gcd (a000203 n) n  -- Reinhard Zumkeller, Mar 23 2013

Table[GCD[n,DivisorSigma[1,n]],{n,110}] (* Harvey P. Dale, Aug 23 2015 *)

a(n) = gcd(n, sigma(n)); \\ Michel Marcus, Oct 23 2013

A000005(a(n)) = A073802(n). - Reinhard Zumkeller, Mar 12 2010
A006530(a(n)) = A082062(n). - Reinhard Zumkeller, Jul 10 2011
a(A014567(n)) = 1; A069059(a(n)) > 1. - Reinhard Zumkeller, Mar 23 2013
a(n) = n/A017666(n). - Antti Karttunen, May 22 2017

A300242 Filter sequence combining gcd(n,sigma(n)) and gcd(n,phi(n)), (A009194 and A009195).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 5, 6, 1, 7, 1, 6, 8, 9, 1, 10, 1, 11, 5, 6, 1, 12, 13, 6, 14, 15, 1, 3, 1, 16, 8, 6, 1, 17, 1, 6, 5, 18, 1, 19, 1, 7, 20, 6, 1, 21, 22, 23, 8, 11, 1, 24, 13, 25, 5, 6, 1, 26, 1, 6, 14, 27, 1, 3, 1, 11, 8, 6, 1, 28, 1, 6, 13, 7, 1, 19, 1, 29, 30, 6, 1, 31, 1, 6, 8, 32, 1, 33, 34, 7, 5, 6, 35, 36, 1, 37, 20, 38, 1, 3, 1, 39, 20

Offset: 1

Antti Karttunen, Mar 02 2018

nonn

Restricted growth sequence transform of P(A009194(n), A009195(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

			a(15) = a(33) (= 8) because A009194(15) = A009194(33) = 3 and A009195(15) = A009195(33) = 1.
a(20) = a(52) (= 11) because A009194(20) = A009194(52) = 2 and A009195(20) = A009195(52) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A009194, A009195.

Cf. also A286591, A300225, A300240, A300241, A300243.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A009194(n) = gcd(n, sigma(n));
A009195(n) = gcd(n, eulerphi(n));
Aux300242(n) = (1/2)*(2 + ((A009194(n)+A009195(n))^2) - A009194(n) - 3*A009195(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300242(n))),"b300242.txt");

A286570 Compound filter (prime signature of n & gcd(n, sigma(n))): a(n) = P(A046523(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 3, 10, 3, 61, 3, 36, 10, 27, 3, 117, 3, 27, 34, 136, 3, 103, 3, 90, 21, 27, 3, 619, 10, 27, 36, 753, 3, 625, 3, 528, 34, 27, 21, 666, 3, 27, 21, 552, 3, 625, 3, 117, 103, 27, 3, 1323, 10, 78, 34, 90, 3, 430, 21, 489, 21, 27, 3, 2545, 3, 27, 78, 2080, 21, 625, 3, 90, 34, 495, 3, 2773, 3, 27, 78, 117, 21, 625, 3, 1224, 136, 27, 3, 3801, 21, 27, 34, 375, 3

Offset: 1

Antti Karttunen, May 26 2017

nonn

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

Cf. A000027, A009194, A046523, A286360, A286571, A286591.

A009194(n) = gcd(n, sigma(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286570(n) = (1/2)*(2 + ((A046523(n)+A009194(n))^2) - A046523(n) - 3*A009194(n));

from sympy import factorint, gcd, divisor_sigma
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a046523(n), gcd(n, divisor_sigma(n))) # Indranil Ghosh, May 26 2017

(define (A286570 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A009194 n)) 2) (- (A046523 n)) (- (* 3 (A009194 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A009194(n))^2) - A046523(n) - 3*A009194(n)).

cp's OEIS Frontend

A009194 a(n) = gcd(n, sigma(n)).

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A300242 Filter sequence combining gcd(n,sigma(n)) and gcd(n,phi(n)), (A009194 and A009195).

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A286570 Compound filter (prime signature of n & gcd(n, sigma(n))): a(n) = P(A046523(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.

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