A369245 - OEIS

%I #28 Jan 23 2024 16:41:00

%S 0,0,0,0,1,2,2,1,0,0,1,1,0,1,2,0,1

%N 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).

%C 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.

%C 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.

%C 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.

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

%e 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.

%e a(17) >= 1 because there exists (at least one) solution k = 4903038892893242229501 = 3 * 17 * 96138017507710631951 with A003415(k) = 1+A002110(17).

%e For other cases, see examples in A369246.

%o (PARI)

%o \\ Needs also program from A369054.

%o A002110(n) = prod(i=1,n,prime(i));

%o A369245(n) = A369054(A002110(n)+1);

%o (PARI)

%o \\ Optimized version of above, employs the fact that solutions must all be multiples of 3. Outputs also terms for A369246.

%o 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)));

%o A002110(n) = prod(i=1,n,prime(i));

%o A369245(n) = search_for_3k1_cases(A002110(n)+1);

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

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

%K nonn,hard,more

%O 0,6

%A _Antti Karttunen_, Jan 22 2024