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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 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

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Last modified March 26 21:04 EDT 2023. Contains 361552 sequences. (Running on oeis4.)