A369452 - OEIS

%I #11 Jan 24 2024 13:56:08

%S 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,

%T 28,30,32,34,35,39,40,43,43,46,47,49,51,54,54,56,57,65,66,67,68,72,74,

%U 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

%N 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.

%C 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).

%H Antti Karttunen, <a href="/A369452/b369452.txt">Table of n, a(n) for n = 1..20000</a>

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

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

%o (PARI)

%o up_to = 1024; \\ 2*(10^4);

%o 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])));

%o A369462(n) = A369054((12*n)-1);

%o A369452list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369462(n); v[n] = s); (v); };

%o v369452 = A369452list(up_to);

%o A369452(n) = v369452[n];

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

%K nonn

%O 1,8

%A _Antti Karttunen_, Jan 24 2024