A369239 Number of integers whose arithmetic derivative is larger than 1 and equal to the n-th partial sum of primorial numbers.

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

Offset: 1

Author

Antti Karttunen, Jan 18 2024

Keywords

• nonn
• hard
• more

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].

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.
		

Links

• Antti Karttunen, PARI program for computing terms of A351029, A369000, A369239 and related sequences.

Crossrefs

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

Cf. also A351029, A369000.

Programs

PARI

\\ See the attached program.