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 A232172 Partial sums of second arithmetic derivative of natural numbers. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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: A152465 A082968 A136013 * A114884 A132704 A202151 Adjacent sequences:  A232169 A232170 A232171 * A232173 A232174 A232175 KEYWORD nonn AUTHOR Giorgio Balzarotti, Nov 19 2013 STATUS approved

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Last modified February 20 13:33 EST 2019. Contains 320327 sequences. (Running on oeis4.)