A063906 -id:A063906 - 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

A305616 Near 2-hyperperfect numbers: numbers k such that sigma(k) - 3*k/2 - 1/2 is a proper divisor of k.

Original entry on oeis.org

15, 63, 147, 171, 207, 627, 663, 1023, 1647, 1971, 2975, 6399, 18063, 19359, 27639, 40215, 48895, 58563, 78819, 95511, 114231, 133595, 134871, 145915, 147455, 163539, 168507, 172287, 188067, 529983, 680859, 795639, 1207359, 1238571, 1553499, 1588491, 2049219

Offset: 1

Amiram Eldar, Jun 06 2018

nonn

Supersequence of A063906.

A combination of the notions of 2-hyperperfect numbers (A007593) and near-perfect numbers (A181595).

			15 is in the sequence since sigma(15) = 24 and 24 - 3*15/2 - 1/2 = 1 is a proper divisor of 15.

Amiram Eldar, Table of n, a(n) for n = 1..70
Bhabesh Das and Helen K. Saikia, Identities for Near and Deficient Hyperperfect Numbers, Indian Journal in Number Theory, Vol. 3 (2016), pp. 124-134.

Cf. A000203, A007593, A063906, A181595, A305617.

aQ[n_] := Module[{d=DivisorSigma[1, n]-3n/2-1/2}, d>0 && d!=n && IntegerQ[d] && Divisible[n,d]]; Select[Range[1000000], aQ]

isok(n) = (n % 2) && (k = sigma(n) - (3*n+1)/2) && (k>0) && !(n % k) && (k != n); \\ Michel Marcus, Jun 07 2018

A294149 Numbers k such that the sum of divisors of k is divisible by the sum of nontrivial divisors of k (that is, excluding 1 and k).

Original entry on oeis.org

15, 20, 35, 95, 104, 119, 143, 207, 209, 287, 319, 323, 377, 464, 527, 559, 650, 779, 899, 923, 989, 1007, 1023, 1189, 1199, 1343, 1349, 1519, 1763, 1919, 1952, 2015, 2159, 2507, 2759, 2911, 2915, 2975, 3239, 3599, 3827, 4031, 4199, 4607, 5183, 5207, 5249

Offset: 1

Zdenek Cervenka, Oct 23 2017

nonn

Numbers k such that sigma(k)/(sigma(k)-k-1) is a positive integer.

			15 is in the sequence since sigma(15)/(sigma(15)-15-1) = 24/8 = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002808 (composite numbers).
Cf. A000203, A048050.
Cf. A088831 (k=2), A063906 (k=3).

Quiet@ Select[Range[2, 5300], And[IntegerQ[#], # > 1] &[#2/(#2 - #1 - 1)] & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Oct 24 2017 *)

lista(nn) = forcomposite(n=1, nn, if (denominator(sigma(n)/(sigma(n)-n-1)) == 1, print1(n, ", "))); \\ Michel Marcus, Oct 24 2017

list(lim)=my(v=List(),s,t); forfactored(n=9,lim\1, s=sigma(n); t=s-n[1]-1; if(t && s%t==0, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 11 2017

This sequence gives all numbers a(n) in increasing order which satisfy A000203(a(n))/A048050(a(n)) = A000203(a(n))/(A000203(a(n)) - (a(n)+1)) = k(n), with a positive integer k(n) for n >= 1. - Wolfdieter Lang, Nov 10 2017

Edited by Wolfdieter Lang, Nov 10 2017

Name corrected by Michel Marcus, Nov 12 2017

A359625 Least number m such that denominator(sigma(m)/(m+1)) = n, or zero if no such m exists.

Original entry on oeis.org

2, 1, 8, 95, 4, 143, 6, 63, 26, 9, 10, 16415, 12, 111, 44, 255, 16, 273023, 18, 159, 62, 175, 22, 575, 74, 25, 80, 671, 28, 3599, 30, 511, 395, 441, 34, 5183, 36, 303, 116, 8639, 40, 163295, 42, 1055, 134, 101567, 46, 19191876318719, 48, 49, 152, 415, 52, 3887

Offset: 1

Michel Marcus, Jan 07 2023

nonn

First occurrence of n in A339966.

Martin Ehrenstein, Table of n, a(n) for n = 1..10000 (first 1000 terms, except for n = 48, from Michel Marcus)

Cf. A162657 (analog for sigma(m)/m), A339966.

Cf. A063906 (sigma(m)/(m+1) = 3/2)

a(n) = my(k=1); while (denominator(sigma(k)/(k+1)) != n, k++); k;

a(n) = if(n == 1, return(2)); my(k=n-1); while (denominator(sigma(k)/(k+1)) != n, k+=n); k; \\ David A. Corneth, Jan 12 2023

n | (a(n) + 1). - David A. Corneth, Jan 12 2023

a(48)-a(54) from Martin Ehrenstein, Jul 23 2023

A362809 Numbers k for which the area of the first part of the symmetric representation of sigma(k) equals sigma(k)/3 and its width is 1.

Original entry on oeis.org

15, 207, 1023, 2975, 5950, 19359, 147455, 294910, 1207359, 5017599, 2170814463

Offset: 1

Hartmut F. W. Hoft, May 04 2023

The symmetric representation of sigma(k), SRS(k), of every term k in this sequence consists of at least 3 parts, with a(1) = 15 and a(5) = 5950 being the only ones among the first 11 terms for which the SRS consists of exactly 3 parts. A251820 is a subsequence. a(12) > 5*10^9.
Suppose that a(n) = 2^i * q, i >= 0 and q odd. Because the first part of SRS(a(n)) has width 1, the smallest prime factor p of q satisfies p > 2^(i+1) -- see the locations of 1's in the triangle of A237048 and computation of widths in A249223. The area of the first part of SRS(a(n)) is (2^(i+1) - 1) * (q+1)/2 = (a(n) + 2^i) * (2^(i+1) - 1) / 2^(i+1) since the first part has 2^(i+1) - 1 legs (see the formula in A237591). Therefore, when 2^i * q is in the sequence then 2^j * q, for 0 <= j <= i is also. Sigma(a(n)) = A068156(i+1) * (a(n) + 2^i) / 2^(i+1).
Conjecture: The area of the first part of SRS(n) being equal to sigma(n)/3 implies that the first part has width 1.
This is true for all odd a(n) since their first part consists of a single leg of width 1. It also holds for even numbers through 10^7.
Observation: Consider the known 11 terms. Apart from 5950 and 294910 the rest are also in A063906. Question: Is A063906 a subsequence? - Omar E. Pol, Jul 09 2023
Answer: A063906 consists of the odd terms of this sequence since the first part of the symmetric representation of sigma for odd a(n) equals (a(n)+1)/2 which is equivalent to a(n) = 2*sigma(a(n))/3 - 1, i.e., a(n) is in A063906. - Hartmut F. W. Hoft, Jul 10 2023
A063906(10) = 58946212863 is a term here. - Omar E. Pol, Jul 12 2023

			15 belongs to the sequence since SRS(15) consists of the parts {8, 8, 8} of maximum widths {1, 2, 1} and sigma(15) = 24.
294910 belongs to the sequence since SRS(294910) consists of the 5 parts {221184, 109440, 2304, 109440, 221184} of maximum widths {1, 3, 2, 3, 1}, with 109440 + 2304 + 109440 = 211184 and sigma(294910) = 3 * 211184 = 663552.
From _Omar E. Pol_, Jul 07 2023: (Start)
Illustration of a(1) = 15.
The 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8] so the diagram of the symmetric representation of sigma(15) in the fourth quadrant is constructed as shown below:
.                                _
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                          _ _ _|_|
.                      _ _| |      8
.                     |    _|
.                    _|  _|
.                   |_ _|  8
.                   |
.    _ _ _ _ _ _ _ _|
.   |_ _ _ _ _ _ _ _|
.                    8
.
The area of the first part (or polygon) of the diagram equals sigma(15)/3 = 24/3 = 8 and its width is 1 so 15 is in the sequence. (End)

Cf. A068156, A235791, A237048, A237591, A237593, A249223, A251820.

Cf. A063906.

(* substitute code suggested by Andrey Zabolotskiy *)
cd[n_, k_] := Boole[Divisible[n, k]]
a237048[s_, j_] := If[OddQ[j], cd[s, j], cd[s-j/2, j]]
firstZeroQ[s_, a_] := Sum[(-1)^(j+1)a237048[s, j], {j, a}]==0
evenPart[n_] := 2^IntegerExponent[n, 2]
a362809[{m_, n_}] := Module[{a, b}, Select[Range[m, n], (a=evenPart[#]; b=(2a-1)/(2a); DivisorSigma[1, #]==3b(#+a)&&firstZeroQ[#, 2a])&]]
a362809[{1, 2170814463}] (* a(11) has a long computation time *)

A190786 Numbers m such that sigma(2m-1) = 3m, where sigma(k) is the sum of the positive divisors of k.

Original entry on oeis.org

8, 104, 512, 1488, 9680, 73728, 603680, 2508800, 1085407232, 29473106432, 583166845500512, 18236498181611824400

Offset: 1

Luis H. Gallardo, May 19 2011

All even perfect numbers are of the form z(2*z-1) with z = 2^(p-1), p prime and 2*z-1 = 2^p-1 prime. It is unknown if there are any odd perfect numbers of this same form. The equation defining the sequence appears while working a special case of the conjecture.

It is conjectured that all terms of this sequence are even numbers.

			a(1)=8 is a term since sigma(15) = 24 = 3*8.

Cf. A000203, A000396, A063906.

Select[Range[10^5], DivisorSigma[1, 2# - 1] == 3# &] (* Alonso del Arte, May 19 2011 *)

zt(a,b) = {local(c,c1,c2,s); c =a ; c1 = 2*c-1;c2 = 3*c;while(c

a(n) = (A063906(n)+1)/2. - Amiram Eldar, Jan 27 2019

a(9)-a(10) added from the data at A063906 by Amiram Eldar, Jan 27 2019
a(11) from Max Alekseyev, May 22 2025
a(12) from Max Alekseyev, Jul 30 2025

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A305616 Near 2-hyperperfect numbers: numbers k such that sigma(k) - 3*k/2 - 1/2 is a proper divisor of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A294149 Numbers k such that the sum of divisors of k is divisible by the sum of nontrivial divisors of k (that is, excluding 1 and k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A359625 Least number m such that denominator(sigma(m)/(m+1)) = n, or zero if no such m exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A362809 Numbers k for which the area of the first part of the symmetric representation of sigma(k) equals sigma(k)/3 and its width is 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A190786 Numbers m such that sigma(2*m-1) = 3*m, where sigma(k) is the sum of the positive divisors of k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A190786 Numbers m such that sigma(2m-1) = 3m, where sigma(k) is the sum of the positive divisors of k.