A230290 a(n) = Sum_{i=1..n} d(24*i+1) - Sum_{i=1..n} d(6*i+1), where d(n) = A000005(n).

1, 2, 2, 1, 2, 4, 5, 4, 4, 4, 6, 7, 7, 5, 4, 10, 10, 10, 8, 9, 11, 10, 12, 10, 10, 13, 15, 14, 12, 14, 14, 14, 16, 16, 17, 17, 19, 21, 19, 20, 20, 18, 16, 16, 18, 24, 24, 23, 23, 21, 28, 28, 28, 24, 24, 26, 25, 27, 27, 28, 30, 30, 32, 28, 28, 30, 28, 30, 28, 29, 33, 39, 39, 37, 35, 39, 40, 38, 36, 36, 38

Offset: 1

Author

N. J. A. Sloane, Oct 17 2013

Keywords

• nonn

Comments

Every inequality in number theory of the form f(n) >= g(n) is a potential source of a sequence floor(f(n))-ceiling(g(n)).

That sequence can be negative (e.g., floor(2/3)-ceiling(1/3)=-1), but the other 3 differences floor(f(n))-floor(g(n)), ceiling(f(n))-ceiling(g(n)), and ceiling(f(n))-floor(g(n)) are nonnegative. - Jonathan Sondow, Oct 20 2013

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. (Shows that a(n) >= 0.)

Crossrefs

Cf. A000005, A230291, A230292, A230293, A230294, A230295, A230296, A230476.

Programs

Magma

[&+[#Divisors(24*i+1):i in [1..n]] - &+[#Divisors(6*i+1):i in [1..n]]:n in [1..85]]; // Marius A. Burtea, Jan 03 2020

Maple

with(numtheory);
f:=proc(n,a,b,c,d) local i; add(tau(a*i+b),i=1..n) - add(tau(c*i+d),i=1..n); end;
[seq(f(n,24,1,6,1),n=1..120)];
# More efficient:
ListTools:-PartialSums([seq(numtheory:-tau(24*i+1)-numtheory:-tau(6*i+1),i=1..120)]); # Robert Israel, Jan 03 2020

Mathematica

R = Range[100];
Accumulate[DivisorSigma[0, 24R+1] - DivisorSigma[0, 6R+1]] (* Jean-François Alcover, Jan 31 2023 *)

PARI

vector(100, n, sum(i=1, n, numdiv(24*i+1)) - sum(i=1, n, numdiv(6*i+1))) \\ Michel Marcus, Oct 09 2014

Formula

a(n) = (2*log(2)/3) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024