A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).

0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1

Offset: 0

Author

Antti Karttunen, Jan 22 2024

Keywords

• nonn
• hard
• more

Comments

Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum of the form 3*(p+q) + p*q, where p and q are odd primes.

Question: Will there be an eventual growth spurt for this sequence? Even though all solutions must be multiples of 3 (but not of 9), because A006862(n) == 1 (mod 3), for n > 1, and the solutions belong to a set listed by A369461.

Similar sequence A369242 grows more vigorously because A033312(n) == -1 (mod 3) for n >= 3, thus allowing non-multiples of 3 as solutions. See comments in A369252.

Examples

			a(4) = 1 as there exists a natural number 399 = 3 * 7 * 19, whose arithmetic derivative (indicated with 399', see A003415) is computed as ((3*7) + (3*19) + (7*19)) = 211 = 1 + prime(4)# = A006862(4), and because 399 is the unique term in A046316 that satisfies the condition.
a(17) >= 1 because there exists (at least one) solution k = 4903038892893242229501 = 3 * 17 * 96138017507710631951 with A003415(k) = 1+A002110(17).
For other cases, see examples in A369246.
		

Crossrefs

Cf. A002110, A003415, A006862, A046316, A369054, A369246 (the solutions), A369252, A369461.

Cf. also A116979, A369000, A369239 for similar counts, also A369241, A369242 and A369247.

Programs

PARI

\\ Needs also program from A369054.
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = A369054(A002110(n)+1);

PARI

\\ Optimized version of above, employs the fact that solutions must all be multiples of 3. Outputs also terms for A369246.
search_for_3k1_cases(n) = if(3!=(n%4),0, my(p = 5, q, c=0); while(1, q = (n-(3*p)) / (3+p); if(q < p, return(c), if(1==denominator(q) && isprime(q),c++; write("b369246_by_search_order_to.txt", n, " ", 3*p*q))); p = nextprime(1+p)));
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = search_for_3k1_cases(A002110(n)+1);

Formula

a(n) = A369054(A006862(n)).