A370905 Partial sums of the alternating sum of divisors function (A206369).

1, 2, 4, 7, 11, 13, 19, 24, 31, 35, 45, 51, 63, 69, 77, 88, 104, 111, 129, 141, 153, 163, 185, 195, 216, 228, 248, 266, 294, 302, 332, 353, 373, 389, 413, 434, 470, 488, 512, 532, 572, 584, 626, 656, 684, 706, 752, 774, 817, 838, 870, 906, 958, 978, 1018, 1048

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

László Tóth, A survey of the alternating sum-of-divisors function, Acta Universitatis Sapientiae, Mathematica, Vol. 5, No. 1 (2013), pp. 93-107.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Formula

a(n) = Sum_{k=1..n} A206369(k).

a(n) = (Pi^2/30) * n^2 + O(n * log(n)^(2/3) * log(log(n))^(4/3)) (Tóth, 2013).

a(n) = (1/2) * Sum_{k=1..n} A008836(k) * floor(n/k) * floor(n/k + 1). - Daniel Suteu, May 11 2024

Mathematica

f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[beta[#] &, 100]]

Prog

(PARI) beta(n) = {my(f = factor(n)); prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; sum(k = 0, e, (-1)^(e-k)*p^k)); }
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += beta(k); print1(s, ", "))};
(PARI) a(n) = sum(k=1, n, (-1)^bigomega(k) * (n\k) * (n\k+1))/2; \\ Daniel Suteu, May 11 2024
(Python)
from math import prod
from sympy import factorint
def A370905(n): return sum(prod((lambda x:x[0]+int((x[1]<<1)>=p+1))(divmod(p**(e+1), p+1)) for p, e in factorint(k).items()) for k in range(1, n+1)) # Chai Wah Wu, Mar 05 2024

Crossrefs

Cf. A206369, A370906.

Sequence in context: A307207 A165288 A327572 * A362946 A345983 A177754

Adjacent sequences: A370902 A370903 A370904 * A370906 A370907 A370908

Keyword

nonn,easy

Author

Amiram Eldar, Mar 05 2024

Status

approved