A369452 Partial sums of A369462, where A369462(n) = number of representations of 12n-1 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 5, 6, 8, 8, 10, 11, 13, 13, 14, 15, 18, 19, 20, 22, 27, 27, 28, 28, 30, 32, 34, 35, 39, 40, 43, 43, 46, 47, 49, 51, 54, 54, 56, 57, 65, 66, 67, 68, 72, 74, 76, 79, 82, 82, 86, 86, 90, 91, 92, 96, 99, 100, 103, 104, 110, 112, 115, 115, 120, 123, 124, 126, 132, 134, 140, 142, 144, 144

Offset: 1

Author

Antti Karttunen, Jan 24 2024

Keywords

• nonn

Comments

In the case of the numbers of the form 12m+11 (i.e., intersection of numbers of the form 3k+2 with the numbers of the form 4m+3) any such representation must be one of the four most common combinations that p, q and r may obtain mod-3-wise (see the table given in comments of A369252), therefore this sequence grows fastest among A369450(n), A369451(n) and a(n).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A369054, A369057, A369252, A369450, A369451, A369462.

Programs

PARI

up_to = 1024; \\ 2*(10^4);
A369054(n) = if(3!=(n%4),0, my(v = [3,3], ip = #v, r, c=0); while(1, r = (n-(v[1]*v[2])) / (v[1]+v[2]); if(r < v[2], ip--, ip = #v; if(1==denominator(r) && isprime(r),c++)); if(!ip, return(c)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])));
A369462(n) = A369054((12*n)-1);
A369452list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369462(n); v[n] = s); (v); };
v369452 = A369452list(up_to);
A369452(n) = v369452[n];

Formula

a(1) = A369462(1), for n > 1, a(n) = A369462(n) + a(n-1).

(n) = A369057(3*n) - (A369450(n) + A369451(n)).