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A327934 Numbers k such that there is no prime p such that p^p divides k, but for its arithmetic derivative a positive finite number of such primes exist.

%I A327934 #29 Jul 23 2023 22:22:20
%S A327934 15,35,39,51,55,87,91,95,111,115,119,123,143,155,158,159,183,187,203,
%T A327934 215,219,225,235,247,259,267,275,287,291,295,299,303,319,323,327,329,
%U A327934 335,339,355,371,374,391,395,403,407,410,411,415,427,441,447,451,471,473,482,511,515,519,525,527,533,535,543,551,559,579
%N A327934 Numbers k such that there is no prime p such that p^p divides k, but for its arithmetic derivative a positive finite number of such primes exist.
%C A327934 a(15) = 158, being even, is the first term not in A080774 or A202237, while 275 (= 5^2 * 11) is the first odd term not in the latter.
%C A327934 After 1, 3375 (= 3^3 * 5^3) is the first term in A202237 that is not present in this sequence, even though the overwhelming majority of the terms of A202237 are also included here. On the other hand, A080774 is a genuine subsequence of this sequence, as the sum of prime factors of such semiprimes is always a multiple of 4.
%C A327934 In contrast to A360110, this is not a multiplicative semigroup; For example, although 15 = 3*5 and 275 = 5^2 * 11 are both present, their product 15*275 = 4125 = 3 * 5^3 * 11 is not. - _Antti Karttunen_, Jan 31 2023
%H A327934 Antti Karttunen, <a href="/A327934/b327934.txt">Table of n, a(n) for n = 1..20000</a>
%e A327934 From _Antti Karttunen_, Feb 01 2023: (Start)
%e A327934 1 has arithmetic derivative 1' = A003415(1) = 0. There are an infinite number of primes p such that p^p | 0, but because infinity is not a finite number, 1 is excluded from this sequence.
%e A327934 275 = 5^2 * 11 has no divisor of the form p^p, but its arithmetic derivative 275' = 135 = 3^3 * 5 has one divisor of the form p^p, therefore 275 is included in this sequence.
%e A327934 4125 = 3 * 5^3 * 11 has arithmetic derivative 4125' = A003415(4125) = 4225 = 5^2 * 13^2, that has no divisor of the form p^p, therefore 4125 is not included in this sequence.
%e A327934 (End)
%o A327934 (PARI)
%o A327934 A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
%o A327934 A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
%o A327934 A327928(n) = if(n<=1,0,A129251(A003415(n)));
%o A327934 isA327934(n) = ((0==A129251(n))&&(A327928(n)>0));
%o A327934 (PARI) isA327934(n) = A360111(n); \\ _Antti Karttunen_, Feb 01 2023
%Y A327934 Intersection of A048103 and A327929.
%Y A327934 Cf. A003415, A099309, A129251, A202237, A256750, A327932, A359550, A360110, A360111 (characteristic function).
%Y A327934 Cf. A080774 (subsequence).
%K A327934 nonn
%O A327934 1,1
%A A327934 _Antti Karttunen_, Oct 01 2019
%E A327934 Definition clarified to exclude the term 1 by _Antti Karttunen_, Feb 01 2023