A195333 Numbers n that can be expressed as the sum of the arithmetic derivatives of k consecutive numbers starting from n for some k.

1, 2, 4, 6, 25, 27, 33, 42, 221, 274, 581, 1957, 3125, 11406, 47058, 823543, 1535573, 5056941, 19246541, 19571621, 36861842, 50330577, 2590282517, 45546909393

Offset

1,2

Comments

A051674 is a subsequence of this sequence.

Links

Table of n, a(n) for n=1..24.

Formula

n = Sum_{j=1..k} (n+j-1)', for some k >= 1.

Example

k=1: n=27 -> 27 = 27'.
k=2: n=33 -> 33 = 33' + 34' = 14 + 19.
k=3: n=1957 -> 1957 = 1957' + 1958' + 1959' = 122 + 1179 + 656.

Maple

with(numtheory);
A195333:=proc(i)
local b, c, n, p;
for n from 1 to i do c:=0; b:=-1;
  while c<n do b:=b+1; c:=c+(n+b)*add(op(2, p)/op(1, p), p=ifactors(n+b)[2]); od;
  if n=c then print(n); fi; od; end:
A195333(10000000);

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]])]]; 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 *)

Crossrefs

Cf. A003415, A051674, A187807, A216384.

Sequence in context: A261746 A319575 A318609 * A106274 A204661 A284919

Adjacent sequences: A195330 A195331 A195332 * A195334 A195335 A195336

Keyword

nonn,more

Author

Paolo P. Lava, Jan 03 2013

Extensions

a(23)-a(24) from Donovan Johnson, Jan 26 2013

Status

approved