A333171 a(n) = Sum_{k=0..n} d(k^2 + 1), where d(k) is the number of divisors of k (A000005).

1, 3, 5, 9, 11, 15, 17, 23, 27, 31, 33, 37, 41, 49, 51, 55, 57, 65, 71, 75, 77, 85, 89, 97, 99, 103, 105, 113, 117, 121, 125, 133, 139, 147, 151, 155, 157, 165, 171, 175, 177, 183, 187, 199, 203, 207, 211, 227, 231, 235, 239, 243, 247, 255, 257, 265, 267, 283

Offset: 0

Author

Amiram Eldar, Mar 09 2020

Keywords

• nonn

Examples

			a(0) = d(0^1 + 1) = d(1) = 1.
a(1) = d(0^1 + 1) + d(1^1 + 1) = d(1) + d(2) = 1 + 2 = 3.
		

References

• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 166.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Christopher Hooley, On the number of divisors of quadratic polynomials, Acta Mathematica, Vol. 110 (1963), pp. 97-114.
• James McKee, On the average number of divisors of quadratic polynomials, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 117. No. 3 (1995), pp. 389-392, alternative link.
• James McKee, The average number of divisors of an irreducible quadratic polynomial, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 126. No. 1. (1999), pp. 17-22.

Crossrefs

Partial sums of A193432.

Cf. A000005, A002522.

Programs

Mathematica

Accumulate @ Table[DivisorSigma[0, k^2 + 1], {k, 0, 100}]

PARI

a(n) = sum(k=0, n, numdiv(k^2+1)); \\ Michel Marcus, Mar 10 2020

Formula

a(n) ~ (3/Pi) * n * log(n).