|
| |
|
|
A231946
|
|
Partial sums of the third power of the arithmetic derivative function A003415.
|
|
3
|
|
|
|
0, 1, 2, 66, 67, 192, 193, 1921, 2137, 2480, 2481, 6577, 6578, 7307, 7819, 40587, 40588, 49849, 49850, 63674, 64674, 66871, 66872, 152056, 153056, 156431, 176114, 208882, 208883, 238674, 238675, 750675, 753419, 760278, 762006, 978006, 978007, 987268, 991364
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
a(n) grows roughly like 0.66*n^4 as n->oo.
Note: 1^3 + 2^3 + 3^3 + 4^3 + 5^3 ... -> ~ (1/4)*n^4; the asymptotic similarity between the sum of powers of positive integers and the sum of powers of their derivatives stands also with sums in which the terms are higher powers, i.e., Sum_{j=1..n} j'^m -> k*n^(m+1) as Sum_{j=i..n} j^m -> h*n^(m+1) when n->oo, in other words, the ratio of the two sums is a constant.
|
|
|
LINKS
|
Robert Israel, Table of n, a(n) for n = 1..10000
E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., vol. 4, no. 2, May 1961, pp. 117-122.
|
|
|
FORMULA
|
a(n) = Sum_{j=1..n} (j')^3, where j' = A003415(j).
|
|
|
EXAMPLE
|
(1')^3 + (2')^3 + (3')^3 + (4')^3 + (5')^3 = 0+1+1+64+1 = 67, so a(5)=67.
|
|
|
MAPLE
|
der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(i)^3, i=1..j), j=1..60);
|
|
|
MATHEMATICA
|
dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Accumulate[Table[dn[n]^3, {n, 100}]] (* T. D. Noe, Nov 20 2013 *)
|
|
|
CROSSREFS
|
Cf. A003415, A190121, A231864.
Sequence in context: A055765 A265996 A309169 * A333677 A098089 A304934
Adjacent sequences: A231943 A231944 A231945 * A231947 A231948 A231949
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Giorgio Balzarotti, Nov 15 2013
|
|
|
STATUS
|
approved
|
| |
|
|
