A341276 a(n) = 1 + 3*n*(n+1) - Sum_{k=1..n} d(k), where d(k) is the number of divisors of k.

1, 6, 16, 32, 53, 81, 113, 153, 197, 248, 304, 368, 434, 510, 590, 676, 767, 867, 969, 1081, 1195, 1317, 1445, 1581, 1717, 1864, 2016, 2174, 2336, 2508, 2680, 2864, 3050, 3244, 3444, 3650, 3857, 4077, 4301, 4531, 4763, 5007, 5251, 5507, 5765

Offset: 0

Author

Luc Rousseau, Feb 08 2021

Keywords

• nonn

Comments

Number of regions of the plane delimited by the 3n curves: x=k, y=k, x*y=k for k=1..n with x>0 and y>0.

Examples

			The curves with equations x=1, y=1 and x*y=1 are delimiting a(1)=6 regions in the (+; +) quadrant of the plane. With the addition of the curves with equations x=2, y=2 and x*y=2, the number of delimited regions reaches a(2)=16.
		

Crossrefs

Cf. A000005, A006218, A028896.

Programs

Mathematica

Table[1+3n(n+1)-Sum[DivisorSigma[0,k],{k,n}],{n,0,44}] (* Stefano Spezia, Feb 08 2021 *)

PARI

a(n)=1+3*n*(n+1)-sum(k=1,n,n\k)

Python

from sympy import integer_nthroot
def A341276(n): return 1+3*n*(n+1)-2*sum(n//k for k in range(1,integer_nthroot(n,2)[0]+1))+integer_nthroot(n,2)[0]**2 # Chai Wah Wu, Mar 31 2021

Formula

a(n) = 1 + A028896(n) - A006218(n).