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A369239 Number of integers whose arithmetic derivative is larger than 1 and equal to the n-th partial sum of primorial numbers. 8
0, 1, 2, 1, 2, 1, 2, 1, 27, 0, 319, 1 (list; graph; refs; listen; history; text; internal format)
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.
a(13) >= 1 as there are solutions like 5744093403180469, 12538540924097819, etc., probably thousands or even more in total.
a(14) >= 1 [see examples].
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.
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).
For more examples, see A369240.
(PARI) \\ See the attached program.
Cf. also A351029, A369000.
Sequence in context: A295310 A359509 A335665 * A002107 A208845 A232506
Antti Karttunen, Jan 18 2024

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