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%I #12 Apr 06 2019 04:13:56
%S 60,482,3903,5667,175,68951,11039,76939,1157,22746,19847,16195,23806,
%T 345855,824059,269862,2484227,809358,94558183,98337630,6504935
%N Smallest number x such that sigma(x) = sigma(x(n)), where x(n) is the n-th arithmetic derivatives of x and x is not equal to x(n).
%e First 5 arithmetic derivatives of x = 175 are 95, 24, 44, 48, 112 and sigma(175) = sigma(112) = 48.
%e First 9 arithmetic derivatives of x = 1157 are 102, 91, 20, 24, 44, 48, 112, 240, 608 and sigma(1157) = sigma(608) = 1260.
%p with(numtheory); P:= proc(q) local a,b,k,n,p,t;
%p for n from 1 to q do for k from 1 to q do t:=1; b:=k;
%p while t<=n do a:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]); b:=a; t:=t+1; od;
%p if sigma(k)=sigma(a) and k<>a then lprint(n,k); break; fi;
%p od; od; end: P(10^6);
%t d[n_] := If[n < 2, 0, n*Total[#2/#1 & @@@ FactorInteger[n]]]; dd[n_, k_] := Nest [d, n, k]; sigma[x_] := If[x > 0, DivisorSigma[1, x], 0]; aQ[x_, n_] := Module[ {xx = dd[x, n]}, xx != x && sigma[x] == sigma[xx]]; a[n_] := Module[{k = 1}, While[!aQ[k, n], k++]; k]; Array[a, 10] (* _Amiram Eldar_, Apr 06 2019 *)
%Y Cf. A000203, A003415, A246775.
%K nonn,more
%O 1,1
%A _Paolo P. Lava_, Sep 03 2014
%E a(15)-a(21) from _Amiram Eldar_, Apr 06 2019
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