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A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.

%I A255242 #42 Nov 22 2024 16:26:35
%S A255242 0,1,1,4,1,8,1,12,5,10,1,30,1,12,11,32,1,36,1,38,13,16,1,92,7,18,19,
%T A255242 46,1,74,1,80,17,22,15,140,1,24,19,116,1,90,1,62,51,28,1,256,9,62,23,
%U A255242 70,1,136,19,140,25,34,1,286,1,36,61,192,21,122,1,86,29,114
%N A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.
%C A255242 a(n) = 1 if n is prime.
%H A255242 Amiram Eldar, <a href="/A255242/b255242.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Paolo P. Lava)
%H A255242 Jon Maiga, <a href="http://sequencedb.net/s/A255242">Computer-generated formulas for A255242</a>, Sequence Machine.
%F A255242 a(1) = 0.
%F A255242 a(2^k) = k*2^(k-1) = A001787(k), for k>=1.
%F A255242 a(n^k) = (n^k-2^k)/(n-2), for n odd prime and k>=1.
%F A255242 In particular:
%F A255242 a(3^k)  = A001047(k-1);
%F A255242 a(5^k)  = A016127(k-1);
%F A255242 a(7^k)  = A016130(k-1);
%F A255242 a(11^k) = A016135(k-1).
%F A255242 From _Antti Karttunen_, Nov 22 2024: (Start)
%F A255242 a(n) = A330575(n) - n.
%F A255242 Also, following formulas were conjectured by Sequence Machine:
%F A255242 a(n) = (A191161(n)-n)/2.
%F A255242 a(n) = Sum_{d|n} A001065(d)*A074206(n/d). [Compare to _David A. Corneth_'s Apr 13 2020 formula for A330575]
%F A255242 a(n) = Sum_{d|n} A051953(d)*A067824(n/d).
%F A255242 a(n) = Sum_{d|n} A000203(d)*A174726(n/d).
%F A255242 a(n) = Sum_{d|n} A062790(d)*A253249(n/d).
%F A255242 a(n) = Sum_{d|n} A157658(d)*A191161(n/d).
%F A255242 a(n) = Sum_{d|n} A174725(d)*A211779(n/d).
%F A255242 a(n) = Sum_{d|n} A245211(d)*A323910(n/d).
%F A255242 (End)
%e A255242 The aliquot parts of 8 are 1, 2, 4 and their sum is 7.
%e A255242 Now, let us calculate the aliquot parts of 1, 2 and 4:
%e A255242 1 => 0;  2 => 1;  4 => 1, 2.  Their sum is 0 + 1 + 1 + 2 = 4.
%e A255242 Let us calculate the aliquot parts of 1, 1, 2:
%e A255242 1 => 0;  1 = > 0; 2 => 1. Their sum is 1.
%e A255242 We have left 1: 1 => 0.
%e A255242 Finally, 7 + 4 + 1 = 12. Therefore a(8) = 12.
%p A255242 with(numtheory): P:=proc(q) local a,b,c,k,n,t,v;
%p A255242 for n from 1 to q do b:=0; a:=sort([op(divisors(n))]); t:=nops(a)-1;
%p A255242 while add(a[k],k=1..t)>0 do b:=b+add(a[k],k=1..t); v:=[];
%p A255242 for k from 2 to t do c:=sort([op(divisors(a[k]))]); v:=[op(v),op(c[1..nops(c)-1])]; od;
%p A255242 a:=v; t:=nops(a); od; print(b); od; end: P(10^3);
%t A255242 f[s_] := Flatten[Most[Divisors[#]] & /@ s]; a[n_] := Total@Flatten[FixedPointList[ f, {n}]] - n; Array[a, 100] (* _Amiram Eldar_, Apr 06 2019 *)
%o A255242 (PARI) ali(n) = setminus(divisors(n), Set(n));
%o A255242 a(n) = my(list = List(), v = [n]); while (#v, my(w = []); for (i=1, #v, my(s=ali(v[i])); for (j=1, #s, w = concat(w, s[j]); listput(list, s[j]));); v = w;); vecsum(Vec(list)); \\ _Michel Marcus_, Jul 15 2023
%Y A255242 Cf. A001047, A001065, A001787, A006516, A016127, A016130, A016135, A255243, A330575.
%Y A255242 Sequences that appear in the convolution formulas: A000203, A001065, A051953, A062790, A067824, A074206, A157658, A174725, A174726, A191161, A211779, A245211, A323910, A253249.
%K A255242 nonn
%O A255242 1,4
%A A255242 _Paolo P. Lava_, Feb 19 2015