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A328321 Numbers n for which A328311(n) = 1 + A051903(A003415(n)) - A051903(n) is strictly positive.

%I A328321 #24 Nov 23 2019 04:07:05
%S A328321 4,6,10,12,14,15,16,20,21,22,26,27,28,30,33,34,35,36,38,39,42,44,46,
%T A328321 48,50,51,52,54,55,57,58,60,62,64,65,66,68,69,70,74,76,77,78,80,82,84,
%U A328321 85,86,87,91,92,93,94,95,99,100,102,105,106,108,110,111,112,114,115,116,118,119,122,123,124,129,130,132,133
%N A328321 Numbers n for which A328311(n) = 1 + A051903(A003415(n)) - A051903(n) is strictly positive.
%C A328321 Numbers n for which A051903(A003415(n)) >= A051903(n), i.e., numbers such that taking their arithmetic derivative does not decrease their "degree", A051903, the maximal exponent in prime factorization.
%H A328321 Antti Karttunen, <a href="/A328321/b328321.txt">Table of n, a(n) for n = 1..10000</a>
%e A328321 10 = 2*5 has maximal exponent (A051903) 1, and its arithmetic derivative A003415(10) = 2+5 = 7 also has maximal exponent 1, thus 10 is included in this sequence.
%e A328321 15 = 3*5 has maximal exponent 1, and its arithmetic derivative A003415(15) = 3+5 = 8 = 2^3 has maximal exponent 3, thus 15 is included in this sequence.
%e A328321 For 8 = 2^3, its arithmetic derivative A003415(8) = 12 = 2^2 * 3, and as 2 < 3 (highest exponent of 12 is less than that of 8), 8 is NOT included here, and from this we also see that A100716 is not a subsequence of this sequence.
%o A328321 (PARI)
%o A328321 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328321 A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
%o A328321 A328311(n) = if(n<=1,0,1+(A051903(A003415(n)) - A051903(n)));
%o A328321 isA328321(n) = (A328311(n)>0);
%Y A328321 Cf. A003415, A051903, A100716, A328302, A328310, A328311.
%Y A328321 Cf. A328320 (complement), A051674, A157037, A328304, A328305 (subsequences).
%K A328321 nonn
%O A328321 1,1
%A A328321 _Antti Karttunen_, Oct 13 2019