A108775 -id:A108775 - cp's OEIS frontend

A045768 Numbers k such that sigma(k) == 2 (mod k).

1, 20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504

Offset: 1

Dan Hoey

nonn

Equivalently, Chowla function of k is congruent to 1 (mod k).
If p=2^i-3 is prime, then 2^(i-1)*p is a term of the sequence. 650 is in the sequence, but is not of this form.
Terms k from a(2) to a(14) satisfy sigma(k) = 2*k + 2, implying that sigma(k) == 0 (mod k+1). It is not known if this holds in general, for there might be solutions of sigma(k)=3k+2 or 4k+2 or ... (Comments from Jud McCranie and Dean Hickerson, updated by Jon E. Schoenfield, Sep 25 2021 and by Max Alekseyev, May 23 2025).
k | sigma(k) produces the multiperfect numbers (A007691). It is an open question whether k | sigma(k) - 1 iff k is a prime or 1. It is not known if there exist solutions to sigma(k) = 2k+1.
Sequence also gives the nonprime solutions to sigma(k) == 0 (mod k+1), k > 1. - Benoit Cloitre, Feb 05 2002
Sequence seems to give nonprime k such that the numerator of the sum of the reciprocals of the divisors of k equals k+1 (nonprime k such that A017665(k)=k+1). - Benoit Cloitre, Apr 04 2002
For k > 1, composite numbers k such that A108775(k) = floor(sigma(k)/k) = sigma(k) mod k = A054024(k). Complement of primes (A000040) with respect to A230606. There are no numbers k > 2 such that sigma(x) = k*(x+1) has a solution. - Jaroslav Krizek, Dec 05 2013

			sigma(650) = 1302 == 2 (mod 650).

R. K. Guy, Unsolved Problems in Number Theory, B2.

Amitabha Tripathi, A note on products of primes that differ by a fixed integer, Fibonacci Quart. 48 (2010), no. 2, 144-149.

Numbers k such that A054013(k)=1.

Cf. A181597, A050414, A050415, A054024, A045769, A088834, A045770, A076496.

Do[If[Mod[DivisorSigma[1, n]-2, n]==0, Print[n]], {n, 1, 10^8}]
Join[{1}, Select[Range[8000000], Mod[DivisorSigma[1, #], #]==2 &]] (* Vincenzo Librandi, Mar 11 2014 *)

is(n)=sigma(n)%n==2 || n==1 \\ Charles R Greathouse IV, Mar 09 2014

More terms from Jud McCranie, Dec 22 1999.
a(11) from Donovan Johnson, Mar 01 2012
a(12) from Giovanni Resta, Apr 02 2014
a(13) from Jud McCranie, Jun 02 2019
Edited and a(14) from Jon E. Schoenfield confirmed by Max Alekseyev, May 23 2025

A284648 Numerator of sum of reciprocals of all divisors of all positive integers <= n.

Original entry on oeis.org

1, 5, 23, 67, 407, 527, 4169, 9913, 33379, 7583, 89461, 102397, 1408777, 1532329, 8238221, 17872837, 316811189, 343357709, 6768841271, 7257705647, 7612437167, 7993370447, 189434541721, 202820113921, 1047296788661, 1090542483461, 3390610314383, 3551237180783, 105395281238707

Offset: 1

Ilya Gutkovskiy, Mar 31 2017

The value of (1/n)*Sum_{k=1..n} sigma(k)/k approaches Pi^2/6.

			1, 5/2, 23/6, 67/12, 407/60, 527/60, 4169/420, 9913/840, 33379/2520, 7583/504, 89461/5544, 102397/5544, 1408777/72072, 1532329/72072, 8238221/360360, ...

József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer, 2006, Section III.5, p. 82.
Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 99.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of divisors

Cf. A000203, A017665, A017666, A108775, A284650 (denominators).

with(numtheory): seq(numer(add(sigma(k)/k, k=1..n)), n=1..40); # Ridouane Oudra, Jan 21 2024

Table[Numerator[Sum[DivisorSigma[-1, k], {k, 1, n}]], {n, 1, 29}]
Table[Numerator[Sum[DivisorSigma[1, k]/k, {k, 1, n}]], {n, 1, 29}]
nmax = 29; Rest[Numerator[CoefficientList[Series[1/(1 - x) Sum[Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x]]]

for(n=1, 29, print1(numerator(sum(k=1, n, sigma(k)/k)),", ")) \\ Indranil Ghosh, Mar 31 2017

from sympy import divisor_sigma, Integer
print([sum(divisor_sigma(k)/Integer(k) for k in range(1, n + 1)).numerator for n in range(1, 30)]) # Indranil Ghosh, Mar 31 2017

G.f.: (1/(1 - x))*Sum_{k>=1} log(1/(1 - x^k)) (for a(n)/A284650(n), see example).
a(n) = numerator of Sum_{k=1..n} Sum_{d|k} 1/d.
a(n) = numerator of Sum_{k=1..n} sigma(k)/k.
a(n) = numerator of Sum_{k=1..n} floor(n/k)/k. - Ridouane Oudra, Jan 21 2024

A284650 Denominator of sum of reciprocals of all divisors of all positive integers <= n.

Original entry on oeis.org

1, 2, 6, 12, 60, 60, 420, 840, 2520, 504, 5544, 5544, 72072, 72072, 360360, 720720, 12252240, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800

Offset: 1

Ilya Gutkovskiy, Mar 31 2017

			1, 5/2, 23/6, 67/12, 407/60, 527/60, 4169/420, 9913/840, 33379/2520, 7583/504, 89461/5544, 102397/5544, 1408777/72072, 1532329/72072, 8238221/360360, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of divisors

Cf. A000203, A017665, A017666, A108775, A284648 (numerators).

with(numtheory): seq(denom(add(sigma(k)/k, k=1..n)), n=1..40); # Ridouane Oudra, Jan 21 2024

Table[Denominator[Sum[DivisorSigma[-1, k], {k, 1, n}]], {n, 1, 29}]
Table[Denominator[Sum[DivisorSigma[1, k]/k, {k, 1, n}]], {n, 1, 29}]
nmax = 29; Rest[Denominator[CoefficientList[Series[1/(1 - x) Sum[Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x]]]

for(n=1, 29, print1(denominator(sum(k=1, n, sigma(k)/k)),", ")) \\ Indranil Ghosh, Mar 31 2017

from sympy import divisor_sigma, Integer
print([sum(divisor_sigma(k)/Integer(k) for k in range(1, n + 1)).denominator for n in range(1, 30)]) # Indranil Ghosh, Mar 31 2017

G.f.: (1/(1 - x))*Sum_{k>=1} log(1/(1 - x^k)) (for A284648(n)/a(n), see example).
a(n) = denominator of Sum_{k=1..n} Sum_{d|k} 1/d.
a(n) = denominator of Sum_{k=1..n} sigma(k)/k.
a(n) = denominator of Sum_{k=1..n} floor(n/k)/k. - Ridouane Oudra, Jan 21 2024

A230606 Numbers n such that sigma(n) = k*(n+1) for some integer k.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 20, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 104, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Jaroslav Krizek, Nov 29 2013

nonn

Numbers n such that A108775(n) = floor(sigma(n) / n) = sigma(n) mod n = A054024(n).

Union primes (A000040) and composite numbers A045768 (k = 1 for primes p, k = 2 for composite numbers).

			20 is in sequence because sigma(20) = 42 = 2*21.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000203(sigma(n)), A054024 (sigma(n) mod n), A108775.

Cf. A045768 (sigma(n) == 2 (mod n)).

Select[Range[300],Divisible[DivisorSigma[1,#],#+1]&] (* Harvey P. Dale, May 28 2019 *)

Example clarified by Harvey P. Dale, May 28 2019

A230608 Numbers with abundancy 4 <= sigma(n)/n < 5.

Original entry on oeis.org

27720, 30240, 32760, 50400, 55440, 60480, 65520, 75600, 83160, 85680, 90720, 95760, 98280, 100800, 105840, 110880, 115920, 120120, 120960, 128520, 131040, 138600, 141120, 143640, 151200, 163800, 166320, 171360, 176400, 180180, 181440, 184800, 191520, 194040

Offset: 1

Jaroslav Krizek, Nov 29 2013

nonn

A subsequence of A023198 (numbers with abundancy >= 4). It differs from A023198 from a(31093) on: The term A023198(31093) = 122522400 = A023199(5) = A215264(1) is not in this sequence. It excludes all terms of A215264, but also the 5-perfect numbers A046060, which are neither in this sequence nor in A215264. [Corrected by M. F. Hasler, Dec 05 2013]
A108775(a(n)) = 4.
There are 31092 terms less than 122522399. - T. D. Noe, Dec 04 2013

			27720 is in sequence because sigma(27720) / 27720 = 112320 / 27720 = 4.0519....

Jaroslav Krizek, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A000203, A023198, A023199, A108775.
Cf. A005100 (deficient numbers with abundancy 1 <= a < 2),
Cf. A204829 (numbers with abundancy 2 <= a < 3),
Cf. A204828 (abundant numbers with abundancy 3 <= a < 4).
Cf. A215264 (abundant numbers with abundancy > 5).

Select[Range[200000], 4 <= DivisorSigma[1, #]/# < 5 &] (* T. D. Noe, Dec 04 2013 *)

Corrected and edited by M. F. Hasler, Dec 05 2013

A272008 a(n) is the numerator of the fractional part of sigma(n)/n, where sigma(n) is the sum of the divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 4, 4, 1, 1, 1, 5, 3, 15, 1, 1, 1, 1, 11, 7, 1, 1, 6, 8, 13, 0, 1, 2, 1, 31, 5, 10, 13, 19, 1, 11, 17, 1, 1, 2, 1, 10, 11, 13, 1, 7, 8, 43, 7, 23, 1, 2, 17, 1, 23, 16, 1, 4, 1, 17, 41, 63, 19, 2, 1, 29, 9, 2, 1, 17, 1, 20, 49, 16, 19, 2, 1, 13, 40

Offset: 1

Michel Marcus, May 10 2016

a(n) = 0 when n is a multiply-perfect number (A007691).

a(n) = 1 when n is a prime or if n belongs to A215012.

			The sum of divisors of 4 is 7; its abundancy is 7/4 = 1 + 3/4 so a(4) = 3.

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

Cf. A000203, A007691, A017665, A017666, A108775, A215012, A240923, A243473.

f[n_] := Numerator[FractionalPart[DivisorSigma[1, n]/n]]; Array[f, 81] (* Robert G. Wilson v, Nov 24 2016 *)

a(n) = my(ab = sigma(n)/n); numerator(ab) % denominator(ab);

a(n) = A017665(n) mod A017666(n).

A087233 a(n) = floor(sigma(A002110(n))/A002110(n)); integer quotient of divisor-sum of primorial numbers and primorials.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Labos Elemer, Sep 01 2003

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000203, A002110, A054640, A108775.

q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]]; q[0]=1; Table[Floor[DivisorSigma[1, a=q[u]]/q[u]//N], {u, 1, 300}]
seq[nmax_] := Floor[FoldList[Times, 1, 1 + 1/Prime[Range[nmax]]]]; seq[100] (* Amiram Eldar, Aug 10 2024 *)

a(n) = floor(vecprod(apply(x -> 1 + 1/x, primes(n)))); \\ Amiram Eldar, Aug 10 2024

From Amiram Eldar, Aug 10 2024: (Start)
a(n) = A108775(A002110(n)).
a(n) = floor(A054640(n)/A002110(n)).
a(n) = floor(Product_{k=1..n} (1 + 1/prime(k))). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Aug 10 2024

cp's OEIS Frontend

A045768 Numbers k such that sigma(k) == 2 (mod k).

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A284648 Numerator of sum of reciprocals of all divisors of all positive integers <= n.

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A284650 Denominator of sum of reciprocals of all divisors of all positive integers <= n.

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A230606 Numbers n such that sigma(n) = k*(n+1) for some integer k.

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A230608 Numbers with abundancy 4 <= sigma(n)/n < 5.

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A272008 a(n) is the numerator of the fractional part of sigma(n)/n, where sigma(n) is the sum of the divisors of n.

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A087233 a(n) = floor(sigma(A002110(n))/A002110(n)); integer quotient of divisor-sum of primorial numbers and primorials.

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