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A153485 Sum of all aliquot divisors of all positive integers <= n. 14
0, 1, 2, 5, 6, 12, 13, 20, 24, 32, 33, 49, 50, 60, 69, 84, 85, 106, 107, 129, 140, 154, 155, 191, 197, 213, 226, 254, 255, 297, 298, 329, 344, 364, 377, 432, 433, 455, 472, 522, 523, 577, 578, 618, 651, 677, 678, 754, 762, 805, 826 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Partial sums of A001065.

a(n) is also the sum of first n terms of A000203, minus n-th triangular number.

n is prime if and only if a(n) - a(n-1) = 1. - Omar E. Pol, Dec 31 2012

Also the alternating row sums of A236540. - Omar E. Pol, Jun 23 2014

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A024916(n) - A000217(n).

a(n) = A000217(n-1) - A004125(n). - Omar E. Pol, Jan 28 2014

a(n) = A000290(n) - A000203(n) - A024816(n) - A004125(n) = A024816(n+1) - A004125(n+1). - Omar E. Pol, Jun 23 2014

G.f.: (1/(1 - x))*Sum_{k>=1} k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Jan 22 2017

a(n) = Sum_{k=1..n} k * floor((n-k)/k). - Wesley Ivan Hurt, Apr 02 2017

EXAMPLE

Assuming that a(1) = 0, for n = 6 the aliquot divisors of the first six positive integers are [0], [1], [1], [1, 2], [1], [1, 2, 3], so a(6) = 0 + 1 + 1 + 1 + 2 + 1 + 1 + 2 + 3 = 12.

MATHEMATICA

f[n_] := Sum[ DivisorSigma[1, m] - m, {m, n}]; Array[f, 60] (* Robert G. Wilson v, Jun 30 2014 *)

Accumulate@ Table[DivisorSum[n, # &, # < n &], {n, 51}] (* or *)

Table[Sum[k Floor[(n - k)/k], {k, n}], {n, 51}] (* Michael De Vlieger, Apr 02 2017 *)

PROG

(PARI) a(n) = sum(k=1, n, sigma(k)-k); \\ Michel Marcus, Jan 22 2017

CROSSREFS

Cf. A000027, A000203, A000217, A001065, A024916, A048050, A244049.

Sequence in context: A198331 A057518 A289206 * A244048 A309043 A023143

Adjacent sequences:  A153482 A153483 A153484 * A153486 A153487 A153488

KEYWORD

easy,nonn

AUTHOR

Omar E. Pol, Dec 27 2008

EXTENSIONS

Better name from Omar E. Pol, Jan 28 2014, Jun 23 2014

STATUS

approved

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Last modified November 14 19:59 EST 2019. Contains 329128 sequences. (Running on oeis4.)