cp's OEIS Frontend

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

%I A370905 #15 May 11 2024 09:40:48
%S A370905 1,2,4,7,11,13,19,24,31,35,45,51,63,69,77,88,104,111,129,141,153,163,
%T A370905 185,195,216,228,248,266,294,302,332,353,373,389,413,434,470,488,512,
%U A370905 532,572,584,626,656,684,706,752,774,817,838,870,906,958,978,1018,1048
%N A370905 Partial sums of the alternating sum of divisors function (A206369).
%H A370905 Amiram Eldar, <a href="/A370905/b370905.txt">Table of n, a(n) for n = 1..10000</a>
%H A370905 László Tóth, <a href="https://doi.org/10.2478/ausm-2014-0007">A survey of the alternating sum-of-divisors function</a>, Acta Universitatis Sapientiae, Mathematica, Vol. 5, No. 1 (2013), pp. 93-107.
%H A370905 László Tóth, <a href="https://www.emis.de/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
%F A370905 a(n) = Sum_{k=1..n} A206369(k).
%F A370905 a(n) = (Pi^2/30) * n^2 + O(n * log(n)^(2/3) * log(log(n))^(4/3)) (Tóth, 2013).
%F A370905 a(n) = (1/2) * Sum_{k=1..n} A008836(k) * floor(n/k) * floor(n/k + 1). - _Daniel Suteu_, May 11 2024
%t A370905 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]]
%o A370905 (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));}
%o A370905 lista(kmax) = {my(s = 0); for(k = 1, kmax, s += beta(k); print1(s, ", "))};
%o A370905 (PARI) a(n) = sum(k=1, n, (-1)^bigomega(k) * (n\k) * (n\k+1))/2; \\ _Daniel Suteu_, May 11 2024
%o A370905 (Python)
%o A370905 from math import prod
%o A370905 from sympy import factorint
%o A370905 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
%Y A370905 Cf. A206369, A370906.
%K A370905 nonn,easy
%O A370905 1,2
%A A370905 _Amiram Eldar_, Mar 05 2024