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%I A328385 #22 Oct 16 2019 13:18:29
%S A328385 0,1,1,4,1,5,1,12,6,7,1,16,1,9,8,32,1,21,1,24,7,13,1,44,10,8,27,32,1,
%T A328385 31,1,80,9,19,12,96,1,7,16,68,1,41,1,48,39,25,1,608,14,39,20,56,1,81,
%U A328385 16,92,13,31,1,96,1,9,51,640,18,61,1,72,8,59,1,156,1,16,55,80,18,71,1,3424,108,43,1,128,13,45,32,140,1,123,20,96,19
%N A328385 If n is of the form p^p, a(n) = n, otherwise a(n) is the first number found by iterating the map x -> A003415(x) that is different from n and either a prime, or whose degree (A051903) differs from the degree of n.
%H A328385 Antti Karttunen, <a href="/A328385/b328385.txt">Table of n, a(n) for n = 1..16384</a>
%H A328385 Antti Karttunen, <a href="/A328385/a328385.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A328385 a(1) = 0 [as here the degrees of 0 and 1 are considered different].
%F A328385 a(p) = 1 for all primes.
%F A328385 a(A051674(n)) = A051674(n).
%F A328385 a(A157037(n)) = A003415(A157037(n)), a prime.
%F A328385 a(A328252(n)) = A003415(A328252(n)), a squarefree number.
%F A328385 a(n) = A003415^(k)(n), when k = abs(A328384(n)). [Taking the abs(A328384(n))-th arithmetic derivative of n gives a(n)]
%e A328385 For n = 3, 3 is a prime, thus a(3) = 1.
%e A328385 For n = 4, A003415(4) = 4, thus as it is among the fixed points of A003415 and a(4) = 4.
%e A328385 For n = 8 = 2^3, its "degree" is A051903(33) = 3, but A003415(8) = 12 = 2^2 * 3, with degree 2, thus a(8) = 12.
%e A328385 For n = 21 = 3*7, A051903(21) = 1, the first derivative A003415(21) = 10 = 2*5 is of the same degree as A051903(10) = 1, but then continuing, we have A003415(10) = 7, which is a prime, thus a(21) = 7.
%e A328385 For n = 33 = 3*11, A051903(33) = 1, A003415(33) = 14 = 2*7, is of the same degree, but on the second iteration, A003415(14) = 9 = 3^2, with A051903(9) = 2, different from the initial degree, thus a(33) = 9.
%o A328385 (PARI)
%o A328385 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328385 A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
%o A328385 A328385(n) = { my(d=A051903(n), u=A003415(n)); while(u && (u!=n) && !isprime(u) && A051903(u)==d, n = u; u = A003415(u)); (u); };
%Y A328385 Cf. A003415, A051674, A051903, A157037.
%Y A328385 Cf. A328384 (the number of iterations needed to reach such a number).
%Y A328385 Cf. also A327968, A327975, A327977, A328252, A328305, A328310, A328311, A328320, A328321.
%K A328385 nonn
%O A328385 1,4
%A A328385 _Antti Karttunen_, Oct 14 2019

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