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

1,4

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

Formula

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

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

Prog

(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];

Crossrefs

Partial sums of A369460.

Cf. A369054, A369057, A369252, A369451, A369452.

Sequence in context: A242642 A178041 A181805 * A212010 A366418 A328745

Adjacent sequences: A369447 A369448 A369449 * A369451 A369452 A369453

Keyword

nonn

Author

Antti Karttunen, Jan 24 2024

Status

approved