A057020 -id:A057020 - cp's OEIS frontend

A308051 Decimal expansion of lim_{m->oo} (sqrt(log(m))/m^2) Sum_{k=1..m} sigma(k)/d(k), where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

3, 5, 6, 9, 0, 4, 9, 6, 5, 2, 4, 9, 9, 5, 7, 0, 7, 6, 1, 2, 2, 0, 0, 5, 3, 0, 2, 0, 1, 3, 9, 9, 6, 4, 5, 9, 1, 3, 6, 0, 6, 6, 6, 8, 2, 6, 2, 5, 7, 3, 8, 4, 4, 2, 9, 6, 8, 7, 8, 8, 0, 2, 0, 1, 2, 7, 7, 4, 3, 4, 4, 2, 1, 4, 1, 8, 7, 2, 1, 3, 8, 5, 5, 3, 2, 1, 5

Offset: 0

Amiram Eldar, May 10 2019

			0.35690496524995707612200530201399645913606668262573...

V. I. Arnold, Dynamics, Statistics, and Projective Geometry of Galois Fields, Cambridge University Press, Cambridge, 2011, p. 78.

Paul T. Bateman, Paul Erdös, Carl Pomerance, and E. G. Straus, The arithmetic mean of the divisors of an integer, in: Marvin I. Knopp (ed.), Analytic Number Theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol 899, Springer, Berlin, Heidelberg, 1981, pp. 197-220, alternative link.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 162.
Marcin Mazur and Bogdan V. Petrenko, Representations of analytic functions as infinite products and their application to numerical computations, The Ramanujan Journal, Vol. 34, No. 1 (2014), pp. 129-141; arXiv preprint, arXiv:1202.1335 [math.NT], 2012.

Cf. A000005, A000203, A057020, A057021.

$MaxExtraPrecision = 1000; m = 1000; f[x_] := Log[1 + x]/x/Sqrt[1 - x]; c = Rest[CoefficientList[Series[Log[f[x]], {x, 0, m}], x]]; RealDigits[(1/2/ Sqrt[Pi])*Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals (1/(2*sqrt(Pi))) * Product_{p prime} p^(3/2) * log(1 + 1/p) / sqrt(p-1).

A375080 a(n) is the numerator of ( Sum_{d|n} (n - d) )/tau(n).

Original entry on oeis.org

0, 1, 1, 5, 2, 3, 3, 17, 14, 11, 5, 22, 6, 8, 9, 49, 8, 23, 9, 13, 13, 13, 11, 33, 44, 31, 17, 56, 14, 21, 15, 43, 21, 41, 23, 233, 18, 23, 25, 115, 20, 30, 21, 30, 32, 28, 23, 178, 30, 69, 33, 107, 26, 39, 37, 41, 37, 71, 29, 46, 30, 38, 137, 321, 44, 48, 33, 47, 45, 52

Offset: 1

Stefano Spezia, Jul 29 2024

( Sum_{d|n} (n - d) )/tau(n) is the average distance between n and its divisor.

Cf. A000005, A000203, A057020, A057021 (denominator).

a[n_]:=Numerator[n-DivisorSigma[1,n]/DivisorSigma[0,n]];  Array[a,70]

from math import prod
from fractions import Fraction
from sympy import factorint
def A375080(n):
    f = factorint(n).items()
    return (n-Fraction(prod((p**(e+1)-1)//(p-1) for p, e in f),prod(e+1 for p,e in f))).numerator # Chai Wah Wu, Jul 30 2024

a(n) = numerator((n - sigma(n))/tau(n)).
a(n) = numerator(n - A000203(n)/A000005(n)).
a(n) = numerator(n - A057020(n)/A057021(n)).

A346644 Least k >= 1 such that sigma(k)/tau(k) has denominator n or zero if no k exists.

Original entry on oeis.org

1, 2, 4, 8, 16, 450, 64, 128, 36, 162, 1024, 1800, 4096, 1458, 144, 32768, 65536, 54450, 262144, 405000, 576, 118098, 4194304, 28800, 1296, 1062882, 900, 5832, 268435456, 115200, 1073741824, 2147483648, 9216, 86093442, 5184, 217800, 68719476736, 774840978, 102400

Offset: 1

Benoit Cloitre, Jul 26 2021

nonn

Conjecture: k always exists.

Cf. A057020, A057021, A069081.

seq[max_] := Module[{s = Table[0,{max}], c = 0, n = 1}, While[c < max, d = Denominator[DivisorSigma[1,n]/DivisorSigma[0,n]]; If[d <= max && s[[d]] == 0, c++; s[[d]] = n]; n++]; s]; seq[22] (* Amiram Eldar, Jul 26 2021 *)

a(n)=if(n<0, 0, t=1; while(denominator(sigma(t)/numdiv(t))!=n, t++); t)

a(29)-a(36) from Amiram Eldar, Jul 26 2021
a(37) from David A. Corneth, Jul 26 2021
a(38)-a(39) from Jinyuan Wang, Jul 26 2021

A347077 Numbers m such that sigma(m) / tau(m) = sigma(m - 1) / tau(m - 1) + sigma(m + 1) / tau(m + 1).

Original entry on oeis.org

15063, 18519, 49841, 137607, 179943, 203345, 412763, 421307, 517334, 881851, 1102204, 2003233, 2831435, 3869018, 17378593, 76645063, 107594182, 118012619, 190791881, 418588841, 447287713, 475734745, 632799289, 661709127, 664171759, 900701138, 998754443, 1756922665

Offset: 1

Jaroslav Krizek, Aug 15 2021

nonn

Numbers m such that A057020(m) / A057021(m) = A057020(m - 1) / A057021(m - 1) + A057020(m + 1) / A057021(m + 1).

Corresponding values of fractions sigma(m) / tau(m): 5022, 6174, 7128, 45870, 59982, 31008, 111132, 106680, 99636, 220948, 163044, 263160, 449712, 726864, 2278152, ...

			sigma(15063) / tau(15063) = sigma(15062) / tau(15062) + sigma(15064) / tau(15064); 20088 / 4 = 23976 / 8 + 32400 / 16; 5022 = 2997 + 2025.

Cf. A000005 (tau), A000203 (sigma), A057020, A057021.

[m: m in [2..10^6] | (&+Divisors(m) / #Divisors(m)) eq (&+Divisors(m - 1) / #Divisors(m - 1)) + (&+Divisors(m + 1) / #Divisors(m + 1))]

r[n_] := Divide @@ DivisorSigma[{1, 0}, n]; s = {}; r1 = r[1]; r2 = r[2]; Do[r3 = r[n]; If[r2 == r1 + r3, AppendTo[s, n - 1]]; r1 = r2; r2 = r3, {n, 3, 4*10^6}]; s (* Amiram Eldar, Aug 16 2021 *)
Flatten[Position[Partition[Table[DivisorSigma[1,n]/DivisorSigma[0,n],{n,900000}],3,1],?(#[[2]] == #[[1]]+#[[3]]&),1,Heads->False]]+1 (* The program generates the first 10 terms of the sequence. *) (* _Harvey P. Dale, Apr 19 2024 *)

a(16)-a(18) from Jon E. Schoenfield, Aug 15 2021

a(19)-a(28) from Amiram Eldar, Aug 16 2021

cp's OEIS Frontend

A308051 Decimal expansion of lim_{m->oo} (sqrt(log(m))/m^2) Sum_{k=1..m} sigma(k)/d(k), where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

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A375080 a(n) is the numerator of ( Sum_{d|n} (n - d) )/tau(n).

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A346644 Least k >= 1 such that sigma(k)/tau(k) has denominator n or zero if no k exists.

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A347077 Numbers m such that sigma(m) / tau(m) = sigma(m - 1) / tau(m - 1) + sigma(m + 1) / tau(m + 1).

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