cp's OEIS Frontend

A351315 Sum of the 9th powers of the square divisors of n.

%I A351315 #24 Jul 19 2025 12:30:26
%S A351315 1,1,1,262145,1,1,1,262145,387420490,1,1,262145,1,1,1,68719738881,1,
%T A351315 387420490,1,262145,1,1,1,262145,3814697265626,1,387420490,262145,1,1,
%U A351315 1,68719738881,1,1,1,101560344351050,1,1,1,262145,1,1,1,262145,387420490,1,1,68719738881
%N A351315 Sum of the 9th powers of the square divisors of n.
%C A351315 Inverse Möbius transform of n^9 * c(n), where c(n) is the characteristic function of squares (A010052). - _Wesley Ivan Hurt_, Jun 21 2024
%H A351315 Michael De Vlieger, <a href="/A351315/b351315.txt">Table of n, a(n) for n = 1..10000</a>
%F A351315 a(n) = Sum_{d^2|n} (d^2)^9.
%F A351315 Multiplicative with a(p) = (p^(18*(1+floor(e/2))) - 1)/(p^18 - 1). - _Amiram Eldar_, Feb 07 2022
%F A351315 From _Amiram Eldar_, Sep 20 2023: (Start)
%F A351315 Dirichlet g.f.: zeta(s) * zeta(2*s-18).
%F A351315 Sum_{k=1..n} a(k) ~ (zeta(19/2)/19) * n^(19/2). (End)
%F A351315 a(n) = Sum_{d|n} d^9 * c(d), where c = A010052. - _Wesley Ivan Hurt_, Jun 21 2024
%F A351315 a(n) = Sum_{d|n} lambda(d)*d^9*sigma_9(n/d), where lambda = A008836. - _Ridouane Oudra_, Jul 19 2025
%e A351315 a(16) = 68719738881; a(16) = Sum_{d^2|16} (d^2)^9 = (1^2)^9 + (2^2)^9 + (4^2)^9 = 68719738881.
%t A351315 f[p_, e_] := (p^(18*(1 + Floor[e/2])) - 1)/(p^18 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 07 2022 *)
%t A351315 snp[n_]:=Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^9]; Array[snp,50] (* _Harvey P. Dale_, May 25 2025 *)
%Y A351315 Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), this sequence (k=9), A351316 (k=10).
%Y A351315 Cf. A010052, A008836, A013957.
%K A351315 nonn,easy,mult
%O A351315 1,4
%A A351315 _Wesley Ivan Hurt_, Feb 06 2022