A162657 -id:A162657 - cp's OEIS frontend

A215509 Numerator of sigma(n)/n when n belongs to A162657.

1, 3, 4, 7, 6, 13, 8, 15, 13, 21, 12, 31, 14, 31, 26, 31, 18, 49, 20, 63, 32, 45, 24, 65, 31, 49, 40, 57, 30, 91, 32, 63, 52, 63, 48, 91, 38, 75, 56, 93, 42, 127, 44, 93, 121, 93, 48, 127, 57, 93, 80, 105, 54, 121, 72, 127, 80, 105, 60, 217, 62, 127, 104

Offset: 1

Michel Marcus, Aug 14 2012

nonn

When p is prime, A162657(p)=p so a(p)=p+1.

			A162657(2) = 2, so a(2)=numerator(sigma(2)/2)=numerator(3/2)=3.

Michel Marcus, Table of n, a(n) for n = 1..1000

Equal to or greater than A214409.

A359635 a(n) = A162657(n)/n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 8, 3, 1, 1, 26, 1, 24, 1, 4, 1, 3, 1, 2, 1, 7, 1, 6, 1, 1, 3, 2, 1, 1, 1, 4, 1, 2, 1, 32, 1, 4, 18, 8, 1, 4, 1, 1, 9, 2, 1, 3, 1, 8, 1, 2, 1, 156, 1, 32, 1, 1, 1, 312, 1, 2, 3, 14, 1, 26, 1, 2, 1, 4, 1, 8, 1, 2, 1, 2, 1, 21, 1, 4, 3, 4, 1, 792

Offset: 1

Michel Marcus, Jan 09 2023

nonn

A162657(n) is the least k such that denominator(sigma(k)/k) = n, so it is divisible by n.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A162657.

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

A017666 Denominator of sum of reciprocals of divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 6, 19, 10, 21, 11, 23, 2, 25, 13, 27, 1, 29, 5, 31, 32, 11, 17, 35, 36, 37, 19, 39, 4, 41, 7, 43, 11, 15, 23, 47, 12, 49, 50, 17, 26, 53, 9, 55, 7, 57, 29, 59, 5, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35, 71, 24, 73, 37, 75, 19

Offset: 1

N. J. A. Sloane

Sum_{ d divides n } 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
Denominators of coefficients in expansion of Sum_{n >= 1} x^n/(n*(1-x^n)) = Sum_{n >= 1} log(1/(1-x^n)).
Also n/gcd(n, sigma(n)) = n/A009194(n); also n/lcm(all common divisors of n and sigma(n)). Equals 1 if 6,28,120,496,672,8128,..., i.e., if n is from A007691. - Labos Elemer, Aug 14 2002
a(A007691(n)) = 1. - Reinhard Zumkeller, Apr 06 2012
Denominator of sigma(n)/n = A000203(n)/n. a(n) = 1 for numbers n in A007691 (multiply-perfect numbers), a(n) = 2 for numbers n in A159907 (numbers n with half-integral abundancy index), a(n) = 3 for numbers n in A245775, a(n) = n for numbers n in A014567 (numbers n such that n and sigma(n) are relatively prime). See A162657 (n) - the smallest number k such that a(k) = n. - Jaroslav Krizek, Sep 23 2014
For all n, a(n) <= n, and thus records are obtained for terms of A014567. - Michel Marcus, Sep 25 2014
Conjecture: If a(n) is in A005153, then n is in A005153. In particular, if n has dyadic rational abundancy index, i.e., a(n) is in A000079 (such as A007691 and A159907), then n is in A005153. Since every term of A005153 greater than 1 is even, any odd n such that a(n) in A005153 must be in A007691. It is natural to ask if there exists a generalization of the indicator function for A005153, call it m(n), such that m(n) = 1 for n in A005153, 0 < m(n) < 1 otherwise, and m(a(n)) <= m(n) for all n. See also A050972. - Jaycob Coleman, Sep 27 2014

			1, 3/2, 4/3, 7/4, 6/5, 2, 8/7, 15/8, 13/9, 9/5, 12/11, 7/3, 14/13, 12/7, 8/5, 31/16, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 4th formula.

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Abundancy

Cf. A017665, A027750.

import Data.Ratio ((%), denominator)
a017666 = denominator . sum . map (1 %) . a027750_row
-- Reinhard Zumkeller, Apr 06 2012

[Denominator(DivisorSigma(1,n)/n): n in [1..50]]; // G. C. Greubel, Nov 08 2018

with(numtheory): seq(denom(sigma(n)/n), n=1..76) ; # Zerinvary Lajos, Jun 04 2008

Table[Denominator[DivisorSigma[-1, n]], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Denominator[DivisorSigma[1, n]/n], {n, 1, 50}] (* G. C. Greubel, Nov 08 2018 *)

a(n) = denominator(sigma(n)/n); \\ Michel Marcus, Sep 23 2014

from math import gcd
from sympy import divisor_sigma
def A017666(n): return n//gcd(divisor_sigma(n),n) # Chai Wah Wu, Mar 21 2023

More terms from Labos Elemer, Aug 14 2002

A239578 Smallest number m such that the numerator of sigma(m)/m is equal to n, or zero if no such m exists.

Original entry on oeis.org

1, 6, 2, 3, 24, 5, 4, 7, 10, 1080, 35640, 11, 9, 13, 8, 33, 297600, 17, 588, 19, 20, 1782, 1907020800, 23, 216, 45, 34, 78

Offset: 1

Michel Marcus, Mar 21 2014

If n-1 is prime, a(n) = n-1.
a(29) <= 1176249221876579007725568000.
Index of first occurrence of n in A017665. - Michel Marcus, Mar 24 2014

			a(2) = 6 since 6 is the first perfect number, with 2 as the numerator of sigma(6)/6.
a(3) = 2 because sigma(2)/2 = 3/2 and it is the first number that gives this numerator.

Giovanni Resta, Terms a(n) < 10^12, for n <= 1000

Cf. A017665 (numerator of sigma(n)/n), A162657 (similar sequence but related to denominators).

a(n) = {k = 1; while (numerator(sigma(k)/k) != n, k++); k;}

a(23) = 1907020800 confirmed by Giovanni Resta, Mar 21 2014

A262432 Regular triangle read by rows: T(n, k) gives the number of times that the denominator of sigma(x,-1) (A017666) is equal to k when x goes from 1 to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1

Offset: 1

Michel Marcus, Sep 22 2015

The sum of terms of the n-th row is n.
T(n, n) = 1 when n is in A014567.
T(n, n) = 0 when n is in A069059.
T(n, 1) increases when n is a multiperfect number A007691.
For a given k, the first index n for which T(n,k) = 1 is A162657(k).

			The first 6 terms of A017666 are 1, 2, 3, 4, 5, 1 where 1 appears twice, 2 to 5 appear once and 6 is absent; giving the 6th row: 2, 1, 1, 1, 1, 0.
Triangle starts
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 1;
2, 1, 1, 1, 1, 0;
2, 1, 1, 1, 1, 0, 1;
2, 1, 1, 1, 1, 0, 1, 1;
2, 1, 1, 1, 1, 0, 1, 1, 1;
2, 1, 1, 1, 2, 0, 1, 1, 1, 0;
...

Michel Marcus, Table of n, a(n) for n = 1..5050

Cf. A007691, A014567, A017666, A069059, A162657.

Table[Length@ Select[Range@ n, Denominator[DivisorSigma[-1, #]] == k &], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Sep 22 2015 *)

tabl(nn) = {vds = vector(nn, n, denominator(sigma(n,-1))); for (n=1, nn, vin = vector(n, k, vds[k]); rown = vector(n, k, #select(x->x==k, vin)); for(k=1, n, print1(rown[k], ", ")); print(););}

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

A242370 Triangle read by rows: T(n, k) is the smallest x such that the denominator of sigma(x)/x is equal to n and the numerator of sigma(x)/x is congruent to k modulo n.

Original entry on oeis.org

2, 3, 84, 40, 2, 4, 5, 30, 15, 10, 18, 3, 2, 84, 1907020800, 7, 42, 840, 280, 14, 168, 58752, 40, 32640, 2, 96, 4, 8, 540, 54, 3, 9, 117, 84, 135, 252, 20, 5, 238080, 30, 2, 15, 1120, 10, 10080, 11, 66, 1320, 198, 33, 132, 22, 264, 528, 44, 392448, 18, 40, 3

Offset: 2

Michel Marcus, Jun 07 2014

When p is prime T(p, 1) is equal to p.
When n and k are not coprime, T(n, k) = T(n/gcd(n, k), k/gcd(n,k)).
Next term T(12, 5) is <= 212569733376000 with sigma(x)/x = 65/12 and 65 == 5 mod 12.

			T(2, 1) = 2 since sigma(2)/2 = 3/2 has denominator 2 and numerator 3 == 1(mod 2).
T(3, 1) = 3 since sigma(3)/3 = 4/3 has denominator 3 and numerator 4 == 1(mod 3).
T(3, 2) = 84 since sigma(84)/84 = 8/3 has denominator 3 and numerator 8 == 2(mod 3).
Triangle starts:
2,
3, 84,
40, 2, 4,
5, 30, 15, 10,
18, 3, 2, 84, 1907020800,
7, 42, 840, 280, 14, 168,
...

Cf. A017665 and A017666 (sigma(n)/n), A239578 and A162657 (similar sequences with numerators or denominators).

T(k, n) = {for (i=1, 10^10, ab = sigma(i)/i; if ((numerator(ab) % denominator(ab))/denominator(ab) == k/n, return (i)););}

cp's OEIS Frontend

A215509 Numerator of sigma(n)/n when n belongs to A162657.

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A359635 a(n) = A162657(n)/n.

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A017666 Denominator of sum of reciprocals of divisors of n.

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A239578 Smallest number m such that the numerator of sigma(m)/m is equal to n, or zero if no such m exists.

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A262432 Regular triangle read by rows: T(n, k) gives the number of times that the denominator of sigma(x,-1) (A017666) is equal to k when x goes from 1 to n.

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

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A242370 Triangle read by rows: T(n, k) is the smallest x such that the denominator of sigma(x)/x is equal to n and the numerator of sigma(x)/x is congruent to k modulo n.

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