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%I #19 May 09 2021 11:20:30
%S 1,2,4,6,25,27,33,42,221,274,581,1957,3125,11406,47058,823543,1535573,
%T 5056941,19246541,19571621,36861842,50330577,2590282517,45546909393
%N Numbers n that can be expressed as the sum of the arithmetic derivatives of k consecutive numbers starting from n for some k.
%C A051674 is a subsequence of this sequence.
%F n = Sum_{j=1..k} (n+j-1)', for some k >= 1.
%e k=1: n=27 -> 27 = 27'.
%e k=2: n=33 -> 33 = 33' + 34' = 14 + 19.
%e k=3: n=1957 -> 1957 = 1957' + 1958' + 1959' = 122 + 1179 + 656.
%p with(numtheory);
%p A195333:=proc(i)
%p local b,c,n,p;
%p for n from 1 to i do c:=0; b:=-1;
%p while c<n do b:=b+1; c:=c+(n+b)*add(op(2,p)/op(1,p),p=ifactors(n+b)[2]); od;
%p if n=c then print(n); fi; od; end:
%p A195333(10000000);
%t 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]])]]; t = {}; Do[k = n; sm = dn[n]; While[sm < n, k++; sm = sm + dn[k]]; If[sm == n, AppendTo[t, n]], {n, 100000}]; t (* _T. D. Noe_, Jan 04 2013 *)
%Y Cf. A003415, A051674, A187807, A216384.
%K nonn,more
%O 1,2
%A _Paolo P. Lava_, Jan 03 2013
%E a(23)-a(24) from _Donovan Johnson_, Jan 26 2013
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