cp's OEIS Frontend

A369055 Number of representations of 4n-1 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

%I A369055 #63 Feb 16 2025 08:34:06
%S A369055 0,0,0,0,0,0,1,0,0,1,0,0,1,1,0,0,0,1,2,0,0,1,1,1,0,1,0,1,0,2,1,0,1,1,
%T A369055 0,0,2,2,1,0,0,2,0,0,0,1,1,2,2,1,1,0,1,2,0,0,0,3,0,1,0,2,1,2,0,3,1,1,
%U A369055 1,0,0,1,1,0,2,0,0,5,2,0,0,2,1,1,0,2,0,1,1,2,2,0,2,1,0,2,0,3,1,0,0,4,1,0,1
%N A369055 Number of representations of 4n-1 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.
%C A369055 Number of solutions to 4n-1 = x', where x' is the arithmetic derivative of x (A003415), and x is a product of three odd primes, A046316.
%C A369055 The number of 0's in range [1..10^n], for n=1..7 are: 8, 46, 288, 2348, 21330, 206355, 2079925, etc.
%C A369055 Goldbach's conjecture can be expressed by claiming that each even number > 4 is an arithmetic derivative of an odd semiprime, as (p*q)' = p+q, where p and q are odd primes. One way to extend Goldbach's conjecture to three primes involves applying the arithmetic derivative to all possible products of three odd primes (A046316) as: (p*q*r)' = (p*q) + (p*r) + (q*r), and asking, "Onto which subset of natural numbers does this map surjectively?" Clearly, the above formula can only produce numbers of the form 4m+3, and furthermore, an analysis at A369252 shows that the trisections of this sequence have quite different expected values, being on average the highest in the trisection A369462, which gives the number of representations for the numbers of the form 12m+11. This motivates a new kind of Goldbach-3 conjecture: "All numbers of the form 12*m-1, with m large enough, have at least one representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r." Furthermore, empirical data for sequence A369463 suggests that "large enough" in this case might well be 4224080, as 1+(12*4224079) = 50688949 = A369463(285), with the next term of A369463 so far unknown. Similar conjectures can be envisaged for the arithmetic derivatives of products of four or more primes. - _Antti Karttunen_, Jan 25 2024
%H A369055 Antti Karttunen, <a href="/A369055/b369055.txt">Table of n, a(n) for n = 1..100000</a>
%H A369055 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.
%H A369055 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach Conjecture</a>.
%H A369055 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%F A369055 a(n) = A369054(4*n-1).
%F A369055 a(n) = Sum_{i=1..A002620(4*n-1)} A369058(i)*[A003415(i)==4*n-1], where [ ] is the Iverson bracket.
%e A369055 a(7) = 1 because 4*7 - 1 = 27, which can be represented as a sum of the form (p*q) + (p*r) + (q*r), with all three primes p, q and r = 3.
%e A369055 a(19) = 2 because 4*19 - 1 = 75, which can be represented as a sum of the form (p*q) + (p*r) + (q*r) in two ways, with p=3, q=3 and r=11, or with p = q = r = 5.
%e A369055 a(9999995) = 0 because (4*9999995)-1 = 39999979, which cannot be expressed as a sum (p*q) + (p*r) + (q*r) for any three odd primes p, q and r, whether distinct or not.
%o A369055 (PARI)
%o A369055 \\ We iterate over weakly increasing triplets of odd primes:
%o A369055 A369055list(up_to) = { my(v = [3,3,3], ip = #v, d, u = vector(up_to), lim = -1+(4*up_to)); while(1, d = ((v[1]*v[2]) + (v[1]*v[3]) + (v[2]*v[3])); if(d > lim, ip--, ip = #v; u[(d+1)/4]++); if(!ip, return(u)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])); };
%o A369055 v369055 = A369055list(100001);
%o A369055 A369055(n) = v369055[n];
%Y A369055 Cf. A002620, A003415, A046316, A369054, A369056, A369057 (partial sums), A369058, A369064 (record values), A369241, A369245, A369248, A369460, A369461, A369462, A369463.
%Y A369055 Cf. A369460, A369461, A369462 (trisections), A369450, A369451, A369452 (and their partial sums).
%Y A369055 Cf. also A351029, A369239.
%K A369055 nonn
%O A369055 1,19
%A A369055 _Antti Karttunen_, Jan 20 2024