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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
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
Sequence in context: A055765 A265996 A309169 * A333677 A098089 A304934
KEYWORD
nonn
AUTHOR
Giorgio Balzarotti, Nov 15 2013
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)