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A271342 Sum of all even divisors of all positive integers <= n. 3
0, 2, 2, 8, 8, 16, 16, 30, 30, 42, 42, 66, 66, 82, 82, 112, 112, 138, 138, 174, 174, 198, 198, 254, 254, 282, 282, 330, 330, 378, 378, 440, 440, 476, 476, 554, 554, 594, 594, 678, 678, 742, 742, 814, 814, 862, 862, 982, 982, 1044, 1044, 1128, 1128, 1208, 1208, 1320, 1320, 1380, 1380, 1524, 1524, 1588, 1588, 1714, 1714 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is also the sum of all even divisors of all even positive integers <= n.

a(n) is also the total number of parts in all partitions of all positive integers <= n into an even number of equal parts. - Omar E. Pol, Jun 04 2017

The bisection of this sequence equals twice A024916 (see formulas). - Michel Marcus, Dec 14 2017

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

FORMULA

a(1) = 0.

a(n) = 2*A024916((n-1)/2), if n is odd and n > 1.

a(n) = 2*A024916(n/2), if n is even.

a(n) = A024916(n) - A078471(n).

For n > 1, a(2*n + 1) = a(2*n). - David A. Corneth, Jun 06 2017

EXAMPLE

For n = 6 the divisors of all positive integers <= 6 are [1], [1, 2], [1, 3], [1, 2, 4], [1, 5], [1, 2, 3, 6] and the even divisors of all positive integers <= 6 are [2], [2, 4], [2, 6], so a(6) = 2 + 2 + 4 + 2 + 6 = 16. On the other hand the sum of all the divisors of all positive integers <= 6/2 are [1] + [1 + 2] + [1 + 3] = A024916(3) = 8, so a(6) = 2*8 = 16.

For n = 10, (floor(10/2) = 5) numbers have divisor 2, (floor(10/4) = 2) numbers have divisor 4, ..., (floor(10/10) = 1) numbers have divisor 10. Therefore, a(10) = 5 * 2 + 2 * 4 + 1 * 6 + 1 * 8 + 1 * 10 = 42. - David A. Corneth, Jun 06 2017

MATHEMATICA

Accumulate@ Array[DivisorSum[#, # &, EvenQ] &, 65] (* Michael De Vlieger, Jun 06 2017 *)

PROG

(PARI) a(n) = sum(k=1, n, sumdiv(k, d, (1-d%2)*d)); \\ Michel Marcus, Jun 05 2017

(PARI) a(n) = 2 * sum(k=1, n\2, k*(n\(k<<1))) \\ David A. Corneth, Jun 06 2017

CROSSREFS

Cf. A000203, A006128, A024916, A078471, A146076, A271343.

Partial sums of A146076.

Sequence in context: A268342 A058524 A072576 * A060818 A082887 A137583

Adjacent sequences:  A271339 A271340 A271341 * A271343 A271344 A271345

KEYWORD

nonn

AUTHOR

Omar E. Pol, Apr 08 2016

STATUS

approved

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Last modified July 4 13:35 EDT 2020. Contains 335448 sequences. (Running on oeis4.)