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A370906
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Partial alternating sums of the alternating sum of divisors function (A206369).
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2
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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
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OFFSET
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1,3
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LINKS
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FORMULA
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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).
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MATHEMATICA
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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]]
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PROG
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(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
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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