OFFSET
1,1
COMMENTS
tau(n) is the number of positive divisors of n (A000005).
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jaroslav Krizek)
EXAMPLE
2 is a term because Sum_{d|2} tau(d) = Sum_{d|3} tau(d) = 1 + 2 = 3.
MATHEMATICA
Select[Range@ 500, Total@ Map[DivisorSigma[0, #] &, Divisors@ #] == Total@ Map[DivisorSigma[0, #] &, Divisors[# + 1]] &] (* Michael De Vlieger, Dec 25 2016 *)
PROG
(Magma) [n: n in [1..10000] | &+[NumberOfDivisors(d): d in Divisors(n)] eq &+[NumberOfDivisors(d): d in Divisors(n+1)]]
(PARI) sd(n) = sumdiv(n, d, numdiv(d)); \\ A007425
isok(m) = sd(m) == sd(m+1); \\ Michel Marcus, Apr 28 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Dec 25 2016
STATUS
approved