A280074 Numbers k such that Sum_{d|k} tau(d) = Sum_{d|k+1} tau(d).

2, 14, 21, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 253, 285, 296, 298, 301, 302, 326, 332, 334, 375, 381, 387, 393, 394, 429, 434, 445, 446, 453, 481

Offset: 1

Author

Jaroslav Krizek, Dec 25 2016

Keywords

• nonn

Comments

tau(n) is the number of positive divisors of n (A000005).

Numbers k such that A007425(k) = A007425(k+1).

Subsequence of A052213 and A005237.

Sequence is not the same as A052213, first deviation is at a(212): A052213(212) = 2041, a(212) = 2024. Number 2024 is the smallest number n such that A007425(n) = A007425(n+1) with different prime signatures of numbers n and n+1 (2024 = 2^3 * 11 * 23, 2025 = 3^4 * 5^2; A007425(2024) = A007425(2025) = 90).

Sequence of the smallest numbers k such that Sum_{d|k} tau(d) = Sum_{d|k+1} tau(d) = ... = Sum_{d|k+n-1} tau(d) for n>=1: 1, 2, 33, 19940, 204323, 380480345, 440738966073, ...; conjecture: this sequence is different from A034173.

Examples

			2 is a term because Sum_{d|2} tau(d) = Sum_{d|3} tau(d) = 1 + 2 = 3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jaroslav Krizek)

Crossrefs

Cf. A000005, A005237, A007425, A052213.

Programs

Magma

[n: n in [1..10000] | &+[NumberOfDivisors(d): d in Divisors(n)]  eq &+[NumberOfDivisors(d): d in Divisors(n+1)]]

Mathematica

Select[Range@ 500, Total@ Map[DivisorSigma[0, #] &, Divisors@ #] == Total@ Map[DivisorSigma[0, #] &, Divisors[# + 1]] &] (* Michael De Vlieger, Dec 25 2016 *)

PARI

sd(n) = sumdiv(n, d, numdiv(d)); \\ A007425
isok(m) = sd(m) == sd(m+1); \\ Michel Marcus, Apr 28 2020