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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.
6
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
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.
Sequence in context: A242642 A178041 A181805 * A212010 A366418 A328745
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 24 2024
STATUS
approved