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

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

Author

Jaroslav Krizek, Aug 15 2021

Keywords

• nonn

Comments

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, ...

Examples

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

Crossrefs

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

Programs

Magma

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

Mathematica

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 *)

Extensions

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

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