

A369239


Number of integers whose arithmetic derivative is larger than 1 and equal to the nth partial sum of primorial numbers.


8



0, 1, 2, 1, 2, 1, 2, 1, 27, 0, 319, 1
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OFFSET

1,3


COMMENTS

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].


LINKS



EXAMPLE

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 = 133946395968510692 = 13394639596851067 is a prime, we have at least one solution, with A003415(2*13394639596851067) = A003415(26789279193702134) = 2+13394639596851067 = A143293(14).


PROG

(PARI) \\ See the attached program.


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



