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A332569 a(n) = Sum_{k=1..n} floor(n/k) * ceiling(n/k). 1
1, 5, 12, 23, 36, 54, 74, 97, 125, 156, 186, 226, 268, 306, 354, 409, 458, 515, 574, 636, 710, 778, 838, 922, 1013, 1086, 1168, 1264, 1350, 1452, 1556, 1651, 1762, 1864, 1966, 2105, 2234, 2332, 2448, 2594, 2726, 2864, 3004, 3132, 3294, 3444, 3564, 3736, 3917, 4067 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..50.

FORMULA

G.f.: ((1 + x) / (1 - x)) * Sum_{k>=1} x^k / (1 - x^k)^2.

a(n) = 1 + Sum_{k=1..n-1} (sigma(k) + sigma(k+1)) for n > 0.

MATHEMATICA

Table[Sum[Ceiling[n/k] Floor[n/k], {k, 1, n}], {n, 1, 50}]

Table[1 + Sum[DivisorSigma[1, k] + DivisorSigma[1, k + 1], {k, 1, n - 1}], {n, 1, 50}]

nmax = 50; CoefficientList[Series[((1 + x)/(1 - x)) Sum[x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PROG

(MAGMA) [&+[Floor(n/k)*Ceiling(n/k):k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 16 2020

(PARI) a(n) = sum(k=1, n, my(q=n/k); floor(q) * ceil(q)); \\ Michel Marcus, Feb 17 2020

CROSSREFS

Row sums of A085383.

Cf. A000203, A006218, A006590, A024916, A049697, A222548, A332490.

Sequence in context: A298791 A054307 A172295 * A126573 A000327 A220425

Adjacent sequences:  A332566 A332567 A332568 * A332570 A332571 A332572

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 16 2020

STATUS

approved

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Last modified June 5 03:12 EDT 2020. Contains 334828 sequences. (Running on oeis4.)