

A256532


Product of n and the sum of remainders of n mod k, for k = 1, 2, 3, ..., n.


3



0, 0, 3, 4, 20, 18, 56, 64, 108, 130, 242, 204, 364, 434, 540, 576, 867, 846, 1216, 1220, 1470, 1694, 2254, 2040, 2575, 2912, 3375, 3472, 4379, 4140, 5177, 5344, 6072, 6698, 7630, 7128, 8621, 9424, 10491, 10320, 12177, 11928, 13975, 14432, 15255, 16468, 18941, 17952, 20286, 21000, 22899, 23608, 26765, 26568, 29095
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

a(n) is also the volume (or the total number of unit cubes) of a polycube which is a right prism whose base is the symmetric representation of A004125(n).
Note that the union of this right prism and the irregular staircase after nth stage described in A244580 and the irregular stepped pyramid after (n1)th stage described in A245092, form a hexahedron (or cube) of side length n. This comment is represented by the third formula.


LINKS

Indranil Ghosh, Table of n, a(n) for n = 1..10000


FORMULA

a(n) = n * A004125(n).
a(n) = n^3  A256533(n).
a(n) = n^3  A143128(n)  A175254(n1), n > 1.


EXAMPLE

a(5) = 20 because 5 * (0 + 1 + 2 + 1) = 5 * 4 = 20.
a(6) = 18 because 6 * (0 + 0 + 0 + 2 + 1) = 6 * 3 = 18.
a(7) = 56 because 7 * (0 + 1 + 1 + 3 + 2 + 1) = 7 * 8 = 56.


MATHEMATICA

Table[n*Sum[Mod[n, i], {i, 2, n1}], {n, 55}] (* Ivan N. Ianakiev, May 04 2015 *)


PROG

(PARI) vector(50, n, n*sum(k=1, n, n % k)) \\ Michel Marcus, May 05 2015
(Python)
def A256532(n):
....s=0
....for k in range(1, n+1):
........s+=n%k
....return s*n # Indranil Ghosh, Feb 13 2017


CROSSREFS

Cf. A000203, A000578, A004125, A024916, A143128, A175254, A236104, A236112, A237270, A237271, A237593, A244580, A245092, A256533.
Sequence in context: A050214 A256605 A237884 * A051719 A336619 A240970
Adjacent sequences: A256529 A256530 A256531 * A256533 A256534 A256535


KEYWORD

nonn


AUTHOR

Omar E. Pol, May 03 2015


STATUS

approved



