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a(n) = Sum_{k=1..n} floor(n/k) * 5^(k-1).
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%I #33 Jun 27 2024 03:21:09

%S 1,7,33,164,790,3946,19572,97828,488479,2442235,12207861,61039267,

%T 305179893,1525898649,7629414925,38147071306,190734961932,

%U 953674808838,4768372074464,23841860356470,119209292012746,596046459981502,2980232250997128,14901161254984784

%N a(n) = Sum_{k=1..n} floor(n/k) * 5^(k-1).

%C Partial sums of A339685.

%H Seiichi Manyama, <a href="/A344816/b344816.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{k=1..n} Sum_{d|k} 5^(d-1).

%F G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 5*x^k).

%F G.f.: (1/(1 - x)) * Sum_{k>=1} 5^(k-1) * x^k/(1 - x^k).

%F a(n) ~ 5^n / 4. - _Vaclav Kotesovec_, Jun 05 2021

%F a(n) = (1/4) * Sum_{k=1..n} (5^floor(n/k) - 1). - _Ridouane Oudra_, Mar 05 2023

%p seq(add(5^(k-1)*floor(n/k), k=1..n), n=1..60); # _Ridouane Oudra_, Mar 05 2023

%t a[n_] := Sum[5^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* _Amiram Eldar_, May 29 2021 *)

%o (PARI) a(n) = sum(k=1, n, n\k*5^(k-1));

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, 5^(d-1)));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-5*x^k))/(1-x))

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 5^(k-1)*x^k/(1-x^k))/(1-x))

%o (Magma)

%o A344816:= func< n | (&+[Floor(n/k)*5^(k-1): k in [1..n]]) >;

%o [A344816(n): n in [1..40]]; // _G. C. Greubel_, Jun 26 2024

%o (SageMath)

%o def A344816(n): return sum((n//k)*5^(k-1) for k in range(1,n+1))

%o [A344816(n) for n in range(1,41)] # _G. C. Greubel_, Jun 26 2024

%Y Column k=5 of A344821.

%Y Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), this sequence (q=5), A332533 (q=n).

%K nonn

%O 1,2

%A _Seiichi Manyama_, May 29 2021