A369450 Partial sums of A369460, where A369460(n) = number of representations of 12n-9 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

0, 0, 1, 2, 3, 3, 5, 6, 6, 7, 8, 9, 11, 11, 11, 12, 14, 14, 14, 17, 17, 19, 20, 20, 21, 21, 23, 25, 25, 26, 28, 29, 29, 29, 30, 31, 33, 34, 35, 36, 38, 40, 42, 43, 43, 44, 45, 46, 46, 48, 48, 50, 53, 53, 55, 58, 58, 59, 59, 61, 62, 63, 63, 65, 66, 66, 67, 68, 68, 71, 72, 74, 75, 75, 75, 78, 80, 81, 82, 84, 84, 85

Offset: 1

Author

Antti Karttunen, Jan 24 2024

Keywords

• nonn

Comments

In the case of the numbers of the form 12m+3 (i.e., multiples of 3 among the numbers of the form 4m+3) any such representation must either have p = q = 3, or p == q == r == +1 (mod 3), or -1 (mod 3) for all three primes (see the table given in comments of A369252), therefore the cumulative sum here has an intermediate growth among a(n), A369451(n) and A369452(n).

Links

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

Crossrefs

Partial sums of A369460.

Cf. A369054, A369057, A369252, A369451, 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])));
A369460(n) = A369054((12*n)-9);
A369450list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369460(n); v[n] = s); (v); };
v369450 = A369450list(up_to);
A369450(n) = v369450[n];

Formula

a(1) = A369460(1), for n > 1, a(n) = A369460(n) + a(n-1).

a(n) = A369057(3*n) - (A369451(n) + A369452(n)).