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A370905
Partial sums of the alternating sum of divisors function (A206369).
2
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
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
Sequence in context: A307207 A165288 A327572 * A362946 A345983 A177754
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 05 2024
STATUS
approved