A216384 -id:A216384

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A187807

Numbers n that can be expressed as the sum of the arithmetic derivatives of k previous numbers starting from n for some k >= 1.

+10
2

4, 5, 6, 7, 27, 42, 43, 1310, 3125, 47058, 47059, 151747, 192382, 192383, 244419, 257614, 823543, 28170538, 28170539, 36861843, 48647587, 556429758, 2736456639, 26781610526

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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=1310 -> 1310 = 1310'+1309' = 927+383.

k=3: n=43 -> 43 = 43'+42'+41' = 1+41+1.

MAPLE

with(numtheory);

A187807:= proc(i)

local a, b, c, n.p;

for n from 4 to i do

a:=0; b:=-1;

while a<n do b:=b+1; a:=a+(n-b)*add(op(2, p)/op(1, p), p=ifactors(n-b)[2]); od;

if a=n then print(n); fi; od; end:

A187807(100000000);

CROSSREFS

Cf. A003415, A051674, A195333, A216384.

KEYWORD

nonn,more

AUTHOR

Paolo P. Lava, Jan 07 2013

EXTENSIONS

a(22)-a(24) from Donovan Johnson, Jan 26 2013

STATUS

approved

A195333

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

+10
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 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.