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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).

%I A333171 #11 Mar 10 2020 06:30:57
%S A333171 1,3,5,9,11,15,17,23,27,31,33,37,41,49,51,55,57,65,71,75,77,85,89,97,
%T A333171 99,103,105,113,117,121,125,133,139,147,151,155,157,165,171,175,177,
%U A333171 183,187,199,203,207,211,227,231,235,239,243,247,255,257,265,267,283
%N A333171 a(n) = Sum_{k=0..n} d(k^2 + 1), where d(k) is the number of divisors of k (A000005).
%D A333171 Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 166.
%H A333171 Amiram Eldar, <a href="/A333171/b333171.txt">Table of n, a(n) for n = 0..10000</a>
%H A333171 Christopher Hooley, <a href="https://projecteuclid.org/euclid.acta/1485889355">On the number of divisors of quadratic polynomials</a>, Acta Mathematica, Vol. 110 (1963), pp. 97-114.
%H A333171 James McKee, <a href="https://doi.org/10.1017/S0305004100073242">On the average number of divisors of quadratic polynomials</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 117. No. 3 (1995), pp. 389-392, <a href="https://repository.royalholloway.ac.uk/file/b6b98acf-e851-a353-805e-5e5328a14b73/9/paper2.pdf">alternative link</a>.
%H A333171 James McKee, <a href="https://doi.org/10.1017/S030500419800320X">The average number of divisors of an irreducible quadratic polynomial</a>, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 126. No. 1. (1999), pp. 17-22.
%F A333171 a(n) ~ (3/Pi) * n * log(n).
%e A333171 a(0) = d(0^1 + 1) = d(1) = 1.
%e A333171 a(1) = d(0^1 + 1) + d(1^1 + 1) = d(1) + d(2) = 1 + 2 = 3.
%t A333171 Accumulate @ Table[DivisorSigma[0, k^2 + 1], {k, 0, 100}]
%o A333171 (PARI) a(n) = sum(k=0, n, numdiv(k^2+1)); \\ _Michel Marcus_, Mar 10 2020
%Y A333171 Partial sums of A193432.
%Y A333171 Cf. A000005, A002522.
%K A333171 nonn
%O A333171 0,2
%A A333171 _Amiram Eldar_, Mar 09 2020