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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.

%I A369450 #11 Jan 24 2024 13:56:03
%S A369450 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,
%T A369450 25,25,26,28,29,29,29,30,31,33,34,35,36,38,40,42,43,43,44,45,46,46,48,
%U A369450 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
%N 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.
%C A369450 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).
%H A369450 Antti Karttunen, <a href="/A369450/b369450.txt">Table of n, a(n) for n = 1..20000</a>
%F A369450 a(1) = A369460(1), for n > 1, a(n) = A369460(n) + a(n-1).
%F A369450 a(n) = A369057(3*n) - (A369451(n) + A369452(n)).
%o A369450 (PARI)
%o A369450 up_to = 1024; \\ 2*(10^4);
%o A369450 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 A369450 A369460(n) = A369054((12*n)-9);
%o A369450 A369450list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369460(n); v[n] = s); (v); };
%o A369450 v369450 = A369450list(up_to);
%o A369450 A369450(n) = v369450[n];
%Y A369450 Partial sums of A369460.
%Y A369450 Cf. A369054, A369057, A369252, A369451, A369452.
%K A369450 nonn
%O A369450 1,4
%A A369450 _Antti Karttunen_, Jan 24 2024