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A231946 Partial sums of the third power of 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)-> ~ 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 natural numbers and the sum of powers of their derivatives stands also with sums in which the terms are higher powers, i.e. sum(i'^m, i=1..n)-> k*n^(m+1) as sum (i^m,i=1..n)-> h*n^(m+1) when n->oo, in other words the ratio of 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((i')^3,i=1..n), where i'=A003415.

EXAMPLE

(1')^3+(2')^3+(3')^3+(4')^3+(5')^3=0+1+1+64+1=67->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

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Last modified January 26 15:47 EST 2021. Contains 340439 sequences. (Running on oeis4.)