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Numbers k such that Sum_{d|k} tau(d) = Sum_{d|k+1} tau(d).
2

%I #23 Sep 08 2022 08:46:18

%S 2,14,21,33,34,38,44,57,75,85,86,93,94,98,116,118,122,133,135,141,142,

%T 145,147,158,171,177,201,202,205,213,214,217,218,230,244,253,285,296,

%U 298,301,302,326,332,334,375,381,387,393,394,429,434,445,446,453,481

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

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

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

%C Subsequence of A052213 and A005237.

%C 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).

%C 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.

%H Amiram Eldar, <a href="/A280074/b280074.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Jaroslav Krizek)

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

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

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

%o (PARI) sd(n) = sumdiv(n, d, numdiv(d)); \\ A007425

%o isok(m) = sd(m) == sd(m+1); \\ _Michel Marcus_, Apr 28 2020

%Y Cf. A000005, A005237, A007425, A052213.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Dec 25 2016