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

0,2

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.

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Partial sums of A193432.

Cf. A000005, A002522.

Sequence in context: A268174 A166104 A164121 * A078651 A268732 A101114

Adjacent sequences: A333168 A333169 A333170 * A333172 A333173 A333174

Keyword

nonn

Author

Amiram Eldar, Mar 09 2020

Status

approved