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A216384 Numbers that can be expressed as the sum of their first k consecutive arithmetic derivatives for some k > 1. 3
6, 38, 42, 62, 146, 1145, 4214, 15590, 47058, 121935, 464834, 4049465, 4759314, 7756755, 15838490, 18284105, 127137997, 132734042, 141393578, 353493351, 435485242, 470458377, 1056410914 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If k = 1 is allowed, then numbers of the form p^p also occur, where p is prime. These are the terms of A051674. - T. D. Noe, Dec 27 2012

LINKS

Table of n, a(n) for n=1..23.

EXAMPLE

6’=5; 5’=1; 6=5+1  (k=2);

38’=21; 21’=10; 10’=7; 38=21+10+7 (k=3);

42’=41; 41’=1; 42=41+1 (k=2);

62’=33; 33’=14; 14’=9; 9'=6; 62=33+14+9+6 (k=4);

146’=75; 75’=55; 55’=16; 146=75+55+16 (k=3);

1145’=234; 234’=291; 291’=100; 100'=140; 140'=188; 188'=192; 1145=234+291+100+140+188+192 (k=6).

MAPLE

with(numtheory);

A216384:= proc(i)

local a, b, n, p, pfs;

for n from 1 to i do

pfs:=ifactors(n)[2]; a:=n*add(op(2, p)/op(1, p), p=pfs);

if a<n then b:=a;

   while b<n do

     pfs:=ifactors(a)[2]; a:=a*add(op(2, p)/op(1, p), p=pfs);

     if a=0 then break; else b:=b+a; fi;

   od;

   if b=n then print(n); fi; fi; od;

end:

A216384 (10000000);

MATHEMATICA

d[1]=0; d[n_] := n*Total[#2/#1 & @@@ FactorInteger[n]]; seqQ[n_] := Module[{s = 0, k = n}, While[s < n && k > 0, k = d[k]; s += k]; k < n && s == n]; Select[ Range[16000], seqQ] (* Amiram Eldar, Mar 30 2019 *)

PROG

(Python)

from sympy import factorint

A216384 = []

for n in range(1, 10**5):

....ndsum = nd = sum([int(n*e/p) for p, e in factorint(n).items()])

....while ndsum <= n and nd > 1:

........nd = sum([int(nd*e/p) for p, e in factorint(nd).items()])

........ndsum += nd

........if ndsum == n:

............A216384.append(n)

# Chai Wah Wu, Aug 21 2014

(PARI) der(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);

isok(n) = {my(s = 0, kn = n, nb = 0, d); until (s == kn, d = der(n); if (d==0, return(0)); s += d;  if (s > kn, return (0)); n = d; nb++; ); nb > 1; } \\ Michel Marcus, Mar 30 2019

CROSSREFS

Cf. A003415, A051674.

Sequence in context: A243843 A039293 A055713 * A060454 A060452 A229620

Adjacent sequences:  A216381 A216382 A216383 * A216385 A216386 A216387

KEYWORD

nonn,more

AUTHOR

Paolo P. Lava, Sep 06 2012

EXTENSIONS

a(20)-a(23) from Amiram Eldar, Mar 30 2019

STATUS

approved

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Last modified June 15 17:43 EDT 2019. Contains 324142 sequences. (Running on oeis4.)