cp's OEIS Frontend

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.

%I A369451 #13 Jan 24 2024 13:56:12
%S A369451 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,
%T A369451 15,15,15,18,18,18,18,20,21,21,22,23,23,24,24,25,25,25,25,26,26,27,27,
%U A369451 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
%N 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.
%C A369451 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.
%H A369451 Antti Karttunen, <a href="/A369451/b369451.txt">Table of n, a(n) for n = 1..20000</a>
%F A369451 a(1) = A369461(1), for n > 1, a(n) = A369461(n) + a(n-1).
%F A369451 a(n) = A369057(3*n) - (A369450(n) + A369452(n)).
%o A369451 (PARI)
%o A369451 up_to = 1024; \\ 2*(10^4);
%o A369451 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 A369451 A369461(n) = A369054((12*n)-5);
%o A369451 A369451list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369461(n); v[n] = s); (v); };
%o A369451 v369451 = A369451list(up_to);
%o A369451 A369451(n) = v369451[n];
%Y A369451 Partial sums of A369461.
%Y A369451 Cf. A369054, A369057, A369252, A369450, A369452.
%K A369451 nonn
%O A369451 1,8
%A A369451 _Antti Karttunen_, Jan 24 2024