A205523 -id:A205523 - 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

A205524 Numbers n such that gcd(n, sigma(n)) is not equal to sigma(n) mod n.

Original entry on oeis.org

4, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Jaroslav Krizek, Jan 28 2012

nonn

All terms are nonprime numbers. Complement of A205523.

			Number 25 is in sequence because sigma(25)=31, gcd(25,31) = 1, and 31 mod 25 = 6.

Cf. A000203, A000040, A009194, A054024, A205524, A205525.

Select[Range[100], Mod[GCD[#, DivisorSigma[1, #]] - DivisorSigma[1, #], #] > 0 &] (* T. D. Noe, Feb 03 2012 *)

Corrected by T. D. Noe, Feb 03 2012

A205525 Nonprime numbers k such that gcd(k, sigma(k)) == sigma(k) (mod k).

Original entry on oeis.org

1, 6, 12, 18, 20, 24, 28, 40, 56, 88, 104, 120, 180, 196, 224, 234, 240, 360, 368, 420, 464, 496, 540, 600, 650, 672, 780, 992, 1080, 1344, 1504, 1872, 1888, 1890, 1952, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3724, 3744, 4284, 4320, 4680, 5292, 5376, 5624

Offset: 1

Jaroslav Krizek, Jan 28 2012

nonn

Complement of primes (A000040) with respect to A205523.

			24 is in the sequence because gcd(24; sigma(24)=60) = (sigma(24)=60) mod 24 = 12.

Cf. A000203, A000040, A009194, A054024, A205524, A205523.

Select[Range[10000], ! PrimeQ[#] && Mod[GCD[#, DivisorSigma[1, #]] - DivisorSigma[1, #], #] == 0 &] (* T. D. Noe, Feb 03 2012 *)

isok(k) = if (!isprime(k), my(s=sigma(k)); Mod(gcd(k, s), k) == Mod(s, k)); \\ Michel Marcus, Feb 09 2021

Corrected by T. D. Noe, Feb 03 2012

A284082 Smallest positive m such that n divides sigma_m(n) - j where j is some divisor of n, or 0 if no such m exists.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 3, 2, 4, 1, 1, 1, 1, 0, 5, 1, 1, 2, 6, 3, 1, 1, 2, 1, 5, 0, 4, 2, 2, 1, 9, 0, 1, 1, 3, 1, 8, 5, 11, 1, 2, 2, 2, 8, 4, 1, 5, 0, 1, 0, 14, 1, 2, 1, 5, 0, 6, 2, 5, 1, 0, 7, 9, 1, 0, 1, 18, 10, 2, 0, 0, 1, 0, 4, 10, 1, 2, 8

Offset: 1

Juri-Stepan Gerasimov, Mar 19 2017

nonn

If p is a prime, we have a(p) = 1. In general, if n = p^q with p prime, then a(n) <= q. For every prime power p^q < 10^13 it actually holds a(p^q) = q. Is this true for every prime power? - Giovanni Resta, Mar 20 2017
Yes, this is true: sigma[m](p^q) == 1/(1-p^m) (mod p^q); this is never divisible by p, and == 1 (mod p^q) iff m >= q. - Robert Israel, Apr 27 2017
First occurrence of k: 21, 1, 4, 8, 16, 22, 26, 69, 44, 38, 75, 46, 148, 316, 58, 186, ..., . - Robert G. Wilson v, Apr 14 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Integers n such that n divides sigma_k(n) - i, where i is some divisor of n; A205523 (k = 1), A283970 (k = 2).

Cf. A033950.

f:= proc(n) local m,mm,F,S,P,D,M0,M1;
      F:= ifactors(n)[2];
      if nops(F) = 1 then return F[1][2] fi;
      P:= map(t -> t[1]^t[2], F);
      S:= mul(add(t[1]^(i*m),i=0..t[2]),t=F);
      D:= subs(n=0,numtheory:-divisors(n));
      for mm from 1 to ilcm(op(map(numtheory:-phi, P)))+max(seq(t[2],t=F)) do
        if member(subs(m=mm,S) mod n, D) then return mm fi;
      od;
      0
end proc:
map(f, [$1..100]); # Robert Israel, Apr 27 2017

a[n_] := Block[{ds, d=Divisors[n], m=0}, While[m <= 2*n, m++; ds = DivisorSigma[m, n]; If[ Select[d, Mod[ds-#, n] == 0 &, 1] != {}, Break[]]]; If[m > 2*n, 0, m]]; Array[a, 85] (* assuming that sigma(m,n) mod n has a period <= 2*n, Giovanni Resta, Mar 20 2017 *)

a(56) from Giovanni Resta, Mar 20 2017

A283970 Integers m such that m divides sigma_2(m) - k where k is some divisor of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 17, 19, 23, 25, 29, 30, 31, 35, 36, 37, 40, 41, 43, 47, 48, 49, 50, 53, 59, 60, 61, 65, 67, 71, 73, 76, 79, 83, 89, 97, 101, 103, 107, 109, 113, 120, 121, 127, 130, 131, 132, 136, 137, 139, 140, 149, 150, 151, 157, 163, 167, 169, 173, 175, 179, 180

Offset: 1

Juri-Stepan Gerasimov, Mar 18 2017

nonn

			2 is in this sequence because 2 divides A001157(2) - 1 = 5 - 1 = 4.

Supersequence of A166684.

Cf. A001157 (sigma_2(n): sum of squares of divisors of n), A205523 (integers n such that n divides sigma_1(n) - i where i is some divisor of n), A284082.

[[n: k in [1..n] | Denominator(n/k) eq 1 and
Denominator(((DivisorSigma(2, n))-k)/n) eq 1]: n in [1..100]];

Select[Range@ 180, Function[n, Total@ Boole@ Map[Divisible[ DivisorSigma[2, n] - #, n] &, Divisors@ n] > 0]] (* Michael De Vlieger, Mar 19 2017 *)

isok(n) = fordiv(n, d, if (!((sigma(n, 2) - d) % n), return (1))); \\ Michel Marcus, Mar 18 2017

cp's OEIS Frontend

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

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A205524 Numbers n such that gcd(n, sigma(n)) is not equal to sigma(n) mod n.

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A205525 Nonprime numbers k such that gcd(k, sigma(k)) == sigma(k) (mod k).

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A284082 Smallest positive m such that n divides sigma_m(n) - j where j is some divisor of n, or 0 if no such m exists.

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A283970 Integers m such that m divides sigma_2(m) - k where k is some divisor of m.

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