A231946 -id:A231946 - cp's OEIS frontend

A231864 Partial sums of the second power of arithmetic derivative function A003415.

0, 1, 2, 18, 19, 44, 45, 189, 225, 274, 275, 531, 532, 613, 677, 1701, 1702, 2143, 2144, 2720, 2820, 2989, 2990, 4926, 5026, 5251, 5980, 7004, 7005, 7966, 7967, 14367, 14563, 14924, 15068, 18668, 18669, 19110, 19366, 23990, 23991, 25672, 25673, 27977, 29498

Offset: 1

Giorgio Balzarotti, Nov 14 2013

nonn

a(n)-> ~ 0.4*n^3 as n-> oo (note: 1^2+2^2+3^3+4^4+5^4 ...-> ~ 1/3*n^3)

Note: the partial sums of a power of the arithmetic derivatives of the natural numbers tend to infinity as the partial sums of the natural numbers of the same power. In more general sense: sum(D^d(i)^m, i = 1..n) -> k*n^(m+1) as n-> oo where D^d(i) is the derivative of order d th of the natural number i (d may be = 0, i.e. no derivate).

			(1')^2+(2')^2+(3')^2+(4')^2+(5')^2=0+1+1+16+1=19->a(5)=19.

E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., vol. 4, no. 2, May 1961, pp. 117-122.

Cf. A003415, A190121, A231946

der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(i)^2,i=1..j),j=1..45);

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]^2, {n, 100}]] (* T. D. Noe, Nov 20 2013 *)

a(n) = sum((i')^2, i=1..n) where i'=A003415.

A232172 Partial sums of second arithmetic derivative of natural numbers.

Original entry on oeis.org

0, 0, 0, 4, 4, 5, 5, 21, 26, 27, 27, 59, 59, 65, 77, 157, 157, 167, 167, 211, 218, 219, 219, 267, 274, 282, 309, 389, 389, 390, 390, 566, 575, 576, 592, 684, 684, 694, 726, 798, 798, 799, 799, 911, 927, 937, 937, 1177, 1186, 1225, 1249, 1341, 1341, 1449, 1481

Offset: 1

Giorgio Balzarotti, Nov 19 2013

nonn

a(n) = 1''+2''+3''+4''+5''+...+n'' -> ~ constant * n^2 as n -> oo.

Note: a(n) = sum(D^d(i)^m,i=1..n) -> constant * n^(m+1) as n -> oo where D^d(i) is the derivative of order d th of the natural number i (results on arithmetic derivatives descent from Barbeau's paper in References).

			a(5) = 1'' + 2'' + 3'' + 4'' + 5'' = 0+0+0+4+0 = 4.

E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull. vol. 4, no. 2, May 1961.

Cf. A003415, A068346, A190121, A231946.

der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(der(i)),i=1..j),j=1..55);

a(n) = sum(i'', i=1..n), where i'' is the second arithmetic derivative of i (A068346).

cp's OEIS Frontend

A231864 Partial sums of the second power of arithmetic derivative function A003415.

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