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A258142
Consider the unitary 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 iterations reach a sum equal to themselves.
3
6, 21, 60, 85, 90, 261, 976, 2009, 87360, 97273, 4948133, 68353213
OFFSET
1,1
COMMENTS
A002827 is a subset of this sequence.
No more terms below 10^8. - Amiram Eldar, Jan 12 2019
LINKS
Eric Weisstein's World of Mathematics, Unitary Divisor
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Eric Weisstein's World of Mathematics, Unitary Perfect Number
Wikipedia, Unitary divisor
EXAMPLE
Divisors of 85 are 1, 5, 17, 85. Unitary aliquot parts are 1, 5, 17.
We have:
1 + 5 + 17 = 23;
5 + 17 + 23 = 45;
17 + 23 + 45 = 85.
Divisors of 2009 are 1, 7, 41, 49, 287, 2009.
Unitary aliquot parts are 1, 41, 49. We have:
1 + 41 + 49 = 91;
41 + 49 + 91 = 181;
49 + 91 + 181 = 321;
91 + 181 + 321 = 593;
181 + 321 + 593 = 1095;
321 + 593 + 1095 = 2009.
MAPLE
with(numtheory):P:=proc(q, h) local a, b, k, n, t, v; 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 gcd(b[k], n/b[k])=1 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 print(n); fi; fi; fi; od; end: P(10^9, 10000);
MATHEMATICA
aQ[n_] := Module[{s = Most[Select[Divisors[n], GCD[#, n/#] == 1 &]]}, If[Length[s] == 1, False, While[Total[s] < n, AppendTo[s, Total[s]]; s = Rest[s]]; Total[s] == n]]; Select[Range[2, 10^8], aQ] (* Amiram Eldar, Jan 12 2019 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Paolo P. Lava, May 22 2015
EXTENSIONS
a(11)-a(12) from Amiram Eldar, Jan 12 2019
STATUS
approved