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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.

%I A258142 #19 Feb 16 2025 08:33:25
%S A258142 6,21,60,85,90,261,976,2009,87360,97273,4948133,68353213
%N 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.
%C A258142 A002827 is a subset of this sequence.
%C A258142 No more terms below 10^8. - _Amiram Eldar_, Jan 12 2019
%H A258142 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>
%H A258142 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>
%H A258142 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryPerfectNumber.html">Unitary Perfect Number</a>
%H A258142 Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>
%e A258142 Divisors of 85 are 1, 5, 17, 85. Unitary aliquot parts are 1, 5, 17.
%e A258142 We have:
%e A258142 1 + 5 + 17 = 23;
%e A258142 5 + 17 + 23 = 45;
%e A258142 17 + 23 + 45 = 85.
%e A258142 Divisors of 2009 are 1, 7, 41, 49, 287, 2009.
%e A258142 Unitary aliquot parts are 1, 41, 49. We have:
%e A258142 1 + 41 + 49 = 91;
%e A258142 41 + 49 + 91 = 181;
%e A258142 49 + 91 + 181 = 321;
%e A258142 91 + 181 + 321 = 593;
%e A258142 181 + 321 + 593 = 1095;
%e A258142 321 + 593 + 1095 = 2009.
%p A258142 with(numtheory):P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
%p A258142 for n from 1 to q do if not isprime(n) then b:=sort([op(divisors(n))]); a:=[];
%p A258142 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;
%p A258142 a:=sort(a); b:=nops(a); if b>1 then for k from 1 to b do v[k]:=a[k]; od;
%p A258142 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);
%p A258142 od; if v[t]=n then print(n); fi; fi; fi; od; end: P(10^9,10000);
%t A258142 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 *)
%Y A258142 Cf. A002827, A034444, A246544, A247012, A247013, A248134.
%K A258142 nonn,more
%O A258142 1,1
%A A258142 _Paolo P. Lava_, May 22 2015
%E A258142 a(11)-a(12) from _Amiram Eldar_, Jan 12 2019