A369451 Partial sums of A369461, where A369461(n) = number of representations of 12n-5 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 5, 5, 5, 6, 7, 8, 8, 8, 10, 10, 11, 11, 11, 11, 11, 12, 14, 15, 15, 15, 18, 18, 18, 18, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 30, 30, 30, 31, 32, 34, 34, 34, 36, 37, 38, 38, 39, 39, 39, 40, 42, 42, 42, 43, 46, 46, 46, 46, 47, 47, 47, 47, 47, 47, 49, 50, 52

Offset: 1

Author

Antti Karttunen, Jan 24 2024

Keywords

• nonn

Comments

In the case of the numbers of the form 12m+7 (i.e., intersection of numbers of the form 3k+1 with the numbers of the form 4m+3) any such representation must have p = 3 and q > 3 (see the table given in comments of A369252), therefore the cumulative sum here grows slowest among A369450(n), a(n) and A369452(n). Notably, it seems that a(n) < n for all n.

Links

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

Crossrefs

Partial sums of A369461.

Cf. A369054, A369057, A369252, A369450, A369452.

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])));
A369461(n) = A369054((12*n)-5);
A369451list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369461(n); v[n] = s); (v); };
v369451 = A369451list(up_to);
A369451(n) = v369451[n];

Formula

a(1) = A369461(1), for n > 1, a(n) = A369461(n) + a(n-1).

a(n) = A369057(3*n) - (A369450(n) + A369452(n)).