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A369054 Number of representations of n as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

%I A369054 #41 Jan 25 2024 23:19:06
%S A369054 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
%T A369054 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
%U A369054 0,0,0,1,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,1
%N A369054 Number of representations of n as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.
%C A369054 Number of solutions to n = x', where x' is the arithmetic derivative of x (A003415), and x is a product of three odd primes (not all necessarily distinct, A046316).
%C A369054 See the conjecture in A369055.
%H A369054 Antti Karttunen, <a href="/A369054/b369054.txt">Table of n, a(n) for n = 0..50000</a>
%H A369054 Victor Ufnarovski and Bo Ahlander, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.
%F A369054 a(n) = Sum_{i=1..A002620(n)} A369058(i)*[A003415(i)==n], where [ ] is the Iverson bracket.
%F A369054 For n >= 2, a(n) <= A099302(n).
%e A369054 a(27) = 1 as 27 can be expressed in exactly one way in the form (p*q + p*r + q*r), with p, q, r all being 3 in this case, as 27 = (3*3 + 3*3 + 3*3).
%e A369054 a(311) = 5 as 311 = (3*5 + 3*37 + 5*37) = (3*7 + 3*29 + 7*29) = (3*13 + 3*17 + 13*17) = (5*7 + 5*23 + 7*23) = (7*11 + 7*13 + 11*13). Expressed in the terms of arithmetic derivatives, of the A099302(311) = 8 antiderivatives of 311 [366, 430, 494, 555, 609, 663, 805, 1001], only the last five are products of three odd primes: 555 = 3*5*37, 609 = 3*7*29, 663 = 3*13*17, 805 = 5*7*23, 1001 = 7 * 11 * 13.
%o A369054 (PARI)
%o A369054 \\ Use this for building up a list up to a certain n. We iterate over weakly increasing triplets of odd primes:
%o A369054 A369054list(up_to) = { my(v = [3,3,3], ip = #v, d, u = vector(up_to)); while(1, d = ((v[1]*v[2]) + (v[1]*v[3]) + (v[2]*v[3])); if(d > up_to, ip--, ip = #v; u[d]++); if(!ip, return(u)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])); };
%o A369054 v369054 = A369054list(100001);
%o A369054 A369054(n) = if(!n,n,v369054[n]);
%o A369054 (PARI)
%o A369054 \\ Use this for computing the value of arbitrary n. We iterate over weakly increasing pairs of odd primes:
%o A369054 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])));
%Y A369054 Cf. A369055 [quadrisection, a(4n-1)], and its trisections A369460 [= a((12*n)-9)], A369461 [= a((12*n)-5)], A369462 [= a((12*n)-1)].
%Y A369054 Cf. A002620, A003415, A046316, A099302, A369056, A369058, A369252.
%Y A369054 Cf. A369251 (positions of terms > 0), A369464 (positions of 0's).
%Y A369054 Cf. A369063 (positions of records), A369064 (values of records).
%Y A369054 Cf. A369241 [= a(2^n - 1)], A369242 [= a(n!-1)], A369245 [= a(A006862(n))], A369247 [= a(3*A057588(n))].
%K A369054 nonn
%O A369054 0,76
%A A369054 _Antti Karttunen_, Jan 20 2024