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A195333 Numbers n that can be expressed as the sum of the arithmetic derivatives of k consecutive numbers starting from n for some k. 2
1, 2, 4, 6, 25, 27, 33, 42, 221, 274, 581, 1957, 3125, 11406, 47058, 823543, 1535573, 5056941, 19246541, 19571621, 36861842, 50330577, 2590282517, 45546909393 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A051674 is a subset 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, A216384, A187807.

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

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Last modified September 19 13:56 EDT 2020. Contains 337178 sequences. (Running on oeis4.)