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Primitive elements of A235992: numbers k with an even arithmetic derivative that cannot be represented as a product of two smaller such numbers.
4

%I #28 Jan 19 2023 09:35:53

%S 1,4,8,9,12,15,20,21,24,25,28,33,35,39,40,44,49,51,52,55,56,57,65,68,

%T 69,76,77,85,87,88,91,92,93,95,104,111,115,116,119,121,123,124,129,

%U 133,136,141,143,145,148,152,155,159,161,164,169,172,177,183,184,185,187,188,201,203,205,209,212,213

%N Primitive elements of A235992: numbers k with an even arithmetic derivative that cannot be represented as a product of two smaller such numbers.

%F {k | A358680(k)=1 and 0=Sum_{d|k, 1<d<k} A358680(d)*A358680(k/d)}.

%e For 12, A003415(12) = 12' = 16, an even number, but on the other hand, for its factors 2 and 6, neither has even derivative as 2' = 1, 6' = 5, while for its factors 3 and 4 only the other factor has even derivative, as 3' = 1, 4' = 4, so 12 has no nontrivial pair of factors such that both of them would have even arithmetic derivative, and therefore 12 is included in this sequence.

%o (PARI) isA359829(n) = A359828(n);

%Y Setwise difference A235992 \ A359831.

%Y Cf. A003415, A358680, A359828 (characteristic function).

%Y Cf. A046315 (subsequence).

%K nonn

%O 1,2

%A _Antti Karttunen_, Jan 17 2023