A331071 a(n) = Sum_{k <= n} r_2(k)^2*d(k+1), where r_2 = A004018, d = A000005.

1, 33, 65, 65, 97, 353, 353, 353, 401, 465, 593, 593, 593, 849, 849, 849, 881, 1265, 1297, 1297, 1553, 1553, 1553, 1553, 1553, 2129, 2385, 2385, 2385, 2897, 2897, 2897, 2961, 2961, 3217, 3217, 3249, 3505, 3505, 3505, 3633, 4145, 4145, 4145, 4145, 4401, 4401, 4401, 4401, 4497, 5073, 5073, 5201, 5713, 5713

Offset

0,2

Comments

Partial sums of A330574.

References

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 163.

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Karl-Heinz Indlekofer, Eine asymptotische Formel in der Zahlentheorie, (German) Arch. Math. (Basel) 23 (1972), 619-624. MR0318080 (47 #6629).

Formula

a(n) ~ c * n * log(n)^2, where c is a constant. - Amiram Eldar, Mar 05 2020

Mathematica

Accumulate @ Table[SquaresR[2, n]^2 * DivisorSigma[0, n+1], {n, 0, 50}] (* Amiram Eldar, Mar 05 2020 *)

Prog

(Python)
from math import prod
from sympy import factorint
def A331071(n): return 1+sum(prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(i).items())**2*prod(e+1 for e in factorint(i+1).values())<<4 for i in range(1, n+1)) # Chai Wah Wu, May 29 2026

Crossrefs

Cf. A000005, A004018, A330574.

Sequence in context: A053179 A183347 A116350 * A062059 A285317 A198592

Adjacent sequences: A331068 A331069 A331070 * A331072 A331073 A331074

Keyword

nonn

Author

N. J. A. Sloane, Jan 10 2020

Status

approved