A288419 - OEIS

%I #18 Nov 14 2022 01:37:32

%S 1,9,31,73,131,279,351,585,850,1179,1343,2263,2211,3159,4061,4681,

%T 4931,7650,6879,9563,10881,12087,12191,18135,16406,19899,22990,25623,

%U 24419,36549,29823,37449,41633,44379,45981,62050,50691,61911,68541,76635,68963,97929

%N a(n) = Sum_{d|n} d^3*A000593(n/d).

%C Multiplicative because this sequence is the Dirichlet convolution of A000578 and A000593 which are both multiplicative. - _Andrew Howroyd_, Jul 27 2018

%H Seiichi Manyama, <a href="/A288419/b288419.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Nov 13 2022: (Start)

%F a(n) = A027847(n) for odd n.

%F Multiplicative with a(2^e) = (8^(e+1)-1)/7 and a(p^e) = (p^(3*e+5) - (p^2+p+1)*p^(e+1) + p + 1)/((p^3-1)*(p^2-1)) for p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^4, where c = 7*Pi^4*zeta(3)/2880 = (7/32)*zeta(3)*zeta(4) = (7/32) * A183700 = 0.284596... . (End)

%t f[p_, e_] := (p^(3*e+5) - (p^2+p+1)*p^(e+1) + p + 1)/((p^3-1)*(p^2-1)); f[2, e_] := (8^(e+1)-1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 13 2022 *)

%o (PARI) a(n)={sumdiv(n, d, (n/d)^3*sigma(d>>valuation(d,2)))} \\ _Andrew Howroyd_, Jul 27 2018

%Y Cf. A000578, A027847, A183700, A288414.

%Y Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), this sequence (k=3), A288420 (k=4).

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, Jun 09 2017