

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
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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((n1)/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, (1d%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



