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%I #49 Feb 23 2024 07:27:47
%S 1,1,1,4,1,1,1,4,3,1,1,4,1,1,1,16,1,3,1,4,1,1,1,4,5,1,27,4,1,1,1,16,1,
%T 1,1,12,1,1,1,4,1,1,1,4,3,1,1,16,7,5,1,4,1,27,1,4,1,1,1,4,1,1,3,64,1,
%U 1,1,4,1,1,1,12,1,1,5,4,1,1,1,16,27,1,1,4,1,1,1,4,1,3,1,4,1,1,1,16
%N Greatest common divisor of n and its arithmetic derivative.
%C a(n) = 1 iff n is squarefree (A005117), cf. A068328.
%C This sequence is very probably multiplicative. - _Mitch Harris_, Apr 19 2005
%H T. D. Noe, <a href="/A085731/b085731.txt">Table of n, a(n) for n = 1..10000</a>
%H Victor Ufnarovski and Bo Ahlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
%F a(n) = GCD(n, A003415(n)).
%F Multiplicative with a(p^e) = p^e if p divides e; a(p^e) = p^(e-1) otherwise. - _Eric M. Schmidt_, Oct 22 2013
%F From _Antti Karttunen_, Feb 28 2021: (Start)
%F Thus a(A276086(n)) = A328572(n), by the above formula and the fact that A276086 is a permutation of A048103.
%F a(n) = n / A083346(n) = A190116(n) / A086130(n). (End)
%t d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; a[n_] := GCD[n, d[n]]; Table[a[n], {n, 1, 96}] (* _Jean-François Alcover_, Feb 21 2014 *)
%t f[p_, e_] := p^If[Divisible[e, p], e, e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 31 2023 *)
%o (Haskell)
%o a085731 n = gcd n $ a003415 n -- _Reinhard Zumkeller_, May 10 2011
%o (PARI) a(n) = {my(f = factor(n)); for (i=1, #f~, if (f[i,2] % f[i,1], f[i,2]--);); factorback(f);} \\ _Michel Marcus_, Feb 14 2016
%Y Cf. A003415, A005117, A048103, A068328, A083346, A083347, A086130, A129251, A189100, A189036, A189103, A190116, A276086, A327858, A327938, A328572, A340070.
%Y Cf. A072873 (positions of records), A268398 (partial sums).
%K nonn,easy,mult
%O 1,4
%A _Reinhard Zumkeller_, Jul 20 2003
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