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A333171
a(n) = Sum_{k=0..n} d(k^2 + 1), where d(k) is the number of divisors of k (A000005).
1
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
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.
Sequence in context: A268174 A166104 A164121 * A078651 A268732 A101114
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 09 2020
STATUS
approved