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A263344
Consider the abundant aliquot parts, in ascending order, of a composite number. Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to themselves.
0
1700, 5950, 155574, 274550, 300894, 715275, 758625, 1365234, 1404172, 1542500, 1661750, 2095250, 2239750, 2673250, 2962250, 3106750, 3395750, 3829250, 4226625, 4262750, 4407250, 4700619, 5398750, 6371092, 8167635, 8560024, 12305620, 13725855, 15497625, 15586263
OFFSET
1,1
EXAMPLE
Aliquot parts of 1700 are 1, 2, 4, 5, 10, 17, 20, 25, 34, 50, 68, 85, 100, 170, 340, 425, 850. The abundant numbers are 20, 100, 340. Therefore:
20 + 100 + 340 = 460;
100 + 340 + 460 = 900;
340 + 460 + 900 = 1700.
MAPLE
with(numtheory):P:=proc(q, h) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do if not isprime(n) then b:=sort([op(divisors(n))]); a:=[];
for k from 1 to nops(b)-1 do if sigma(b[k])>2*b[k] then a:=[op(a), b[k]]; fi; od;
a:=sort(a); b:=nops(a); if b>1 then for k from 1 to b do v[k]:=a[k]; od;
t:=b+1; v[t]:=add(v[k], k=1..b); while v[t]<n do t:=t+1; v[t]:=add(v[k], k=t-b..t-1);
od; if v[t]=n then lprint(n, a); fi; fi; fi; od; end: P(10^9, 1000);
MATHEMATICA
seqQ[n_] := Module[{d = Select[Most[Divisors[n]], DivisorSigma[1, #] > 2 # &]}, Switch[Length[d], _?(# < 1 &), False, _?(# == 1 &), d[[1]] == n, _, k = 0; While[k < n, k = Total[d]; d = Rest[AppendTo[d, k]]]; k == n]]; seq = {}; Do[ If[seqQ[n], AppendTo[seq, n]], {n, 2, 10^6}]; seq (* Amiram Eldar, Mar 20 2019 *)
CROSSREFS
Cf. A005101 (abundant numbers), A027751 (aliquot parts), A246544, A247012, A258142, A258270.
Sequence in context: A083610 A020405 A091368 * A159464 A166400 A210121
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Oct 15 2015
EXTENSIONS
More terms from Amiram Eldar, Mar 20 2019
STATUS
approved