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 A319086 a(n) = Sum_{k=1..n} k^2*sigma(k), where sigma is A000203. 4
 1, 13, 49, 161, 311, 743, 1135, 2095, 3148, 4948, 6400, 10432, 12798, 17502, 22902, 30838, 36040, 48676, 55896, 72696, 86808, 104232, 116928, 151488, 170863, 199255, 228415, 272319, 297549, 362349, 393101, 457613, 509885, 572309, 631109, 749045, 801067 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS In general, for m>=1, Sum_{k=1..n} k^m * sigma(k) = Sum_{k=1..n} k^(m+1) * (Bernoulli(m+1, floor(1 + n/k)) - Bernoulli(m+1, 0)) / (m+1), where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 08 2018 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 FORMULA a(n) ~ Pi^2 * n^4/24. a(n) = Sum_{k=1..n} ((k/2) * floor(n/k) * floor(1 + n/k))^2. - Daniel Suteu, Nov 07 2018 MATHEMATICA Accumulate[Table[k^2*DivisorSigma[1, k], {k, 1, 50}]] PROG (PARI) a(n) = sum(k=1, n, k^2*sigma(k)); \\ Michel Marcus, Sep 12 2018 (Python) def A319086(n): return sum((k*(m:=n//k)*(m+1)>>1)**2 for k in range(1, n+1)) # Chai Wah Wu, Oct 20 2023 (Python) from math import isqrt def A319086(n): return ((-((s:=isqrt(n))*(s+1))**3*(2*s+1)>>1) + sum(k**2*(q:=n//k)*(q+1)*(2*k*(2*q+1)+3*q*(q+1)) for k in range(1, s+1)))//12 # Chai Wah Wu, Oct 21 2023 CROSSREFS Cf. A000203, A024916, A143128. Sequence in context: A009951 A274974 A251142 * A146287 A289999 A147346 Adjacent sequences: A319083 A319084 A319085 * A319087 A319088 A319089 KEYWORD nonn AUTHOR Vaclav Kotesovec, Sep 10 2018 STATUS approved

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Last modified May 21 15:23 EDT 2024. Contains 372738 sequences. (Running on oeis4.)