A232172 Partial sums of second arithmetic derivative of natural numbers.

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

1,4

Comments

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).

Links

Table of n, a(n) for n=1..55.

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

Formula

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

Example

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

Maple

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);

Crossrefs

Cf. A003415, A068346, A190121, A231946.

Sequence in context: A357414 A082968 A136013 * A114884 A132704 A202151

Adjacent sequences: A232169 A232170 A232171 * A232173 A232174 A232175

Keyword

nonn

Author

Giorgio Balzarotti, Nov 19 2013

Status

approved