A369239 - OEIS

%I #31 Jan 21 2024 18:10:06

%S 0,1,2,1,2,1,2,1,27,0,319,1

%N Number of integers whose arithmetic derivative is larger than 1 and equal to the n-th partial sum of primorial numbers.

%C Note how there are generally less solutions for even n than for odd n. This is explained by the fact that A143293(2n) == 1 (mod 4) and A143293(2n+1) == 3 (mod 4) and the arithmetic derivative A003415 of a product of any three odd primes (A046316) is always of the form 4k+3, therefore the solution set counted by a(2n) does not have any solutions from A046316 that contribute the majority of the solutions counted by a(2n+1). See also A369055.

%C a(13) >= 1 as there are solutions like 5744093403180469, 12538540924097819, etc., probably thousands or even more in total.

%C a(14) >= 1 [see examples].

%H Antti Karttunen, <a href="/A369239/a369239.txt">PARI program for computing terms of A351029, A369000, A369239 and related sequences</a>.

%e a(12) = 1 as there is a unique solution k such that k' = A143293(12) = 7628001653829, that k being 318745032938881 = 71*173*307*1259*67139. It's also the first solution with more than four prime factors.

%e a(14) >= 1, because as A143293(14)-2 = 13394639596851069-2 = 13394639596851067 is a prime, we have at least one solution, with A003415(2*13394639596851067) = A003415(26789279193702134) = 2+13394639596851067 = A143293(14).

%e For more examples, see A369240.

%o (PARI) \\ See the attached program.

%Y Cf. A003415, A046316, A143293, A328243, A369055, A369240.

%Y Cf. also A351029, A369000.

%K nonn,hard,more

%O 1,3

%A _Antti Karttunen_, Jan 18 2024