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URL: https://oeis.org/A370906

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A370906
Partial alternating sums of the alternating sum of divisors function (A206369).
6
1, 0, 2, -1, 3, 1, 7, 2, 9, 5, 15, 9, 21, 15, 23, 12, 28, 21, 39, 27, 39, 29, 51, 41, 62, 50, 70, 52, 80, 72, 102, 81, 101, 85, 109, 88, 124, 106, 130, 110, 150, 138, 180, 150, 178, 156, 202, 180, 223, 202, 234, 198, 250, 230, 270, 240, 276, 248, 306, 282, 342
OFFSET
1,3
LINKS
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} (-1)^(k+1) * A206369(k).
a(n) = (Pi^2/120) * n^2 + O(n * log(n)^(2/3) * log(log(n))^(4/3)) (Tóth, 2017).
MATHEMATICA
f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(# + 1) * 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 += (-1)^(k+1) * beta(k); print1(s, ", "))};
(Python)
from math import prod
from sympy import factorint
def A370906(n): return sum((1 if k&1 else -1)*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
Similar sequences: A068762, A068773, A307704, A357817, A362028.
Sequence in context: A191854 A129646 A277640 * A165401 A319916 A213074
KEYWORD
sign,easy
AUTHOR
Amiram Eldar, Mar 05 2024
STATUS
approved