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A380839 Numerators of J(n) = Product_{p|n, p odd prime} (p - 1)/(p - 2).

%I A380839 #40 May 30 2025 23:15:50
%S A380839 1,1,2,1,4,2,6,1,2,4,10,2,12,6,8,1,16,2,18,4,12,10,22,2,4,12,2,6,28,8,
%T A380839 30,1,20,16,8,2,36,18,24,4,40,12,42,10,8,22,46,2,6,4,32,12,52,2,40,6,
%U A380839 36,28,58,8,60,30,12,1,16,20,66,16,44,8,70,2,72,36
%N A380839 Numerators of J(n) = Product_{p|n, p odd prime} (p - 1)/(p - 2).
%C A380839 This sequence is similar to A173557 but differences occurs for indices n=35,65,70,...
%C A380839 Coefficients J(n)=a(n)/A307410(n) occurs in many formulas on density of primes with gap 2*n.
%C A380839 Sylvester was the first who uses these coefficients at 1871.
%H A380839 Amiram Eldar, <a href="/A380839/b380839.txt">Table of n, a(n) for n = 1..10000</a>
%H A380839 G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70 (e.g. formula 4.11 p.32).
%H A380839 J. J. Sylvester, <a href="https://archive.org/details/rcin.org.pl.WA35_140283_11371-2_Art109_115452/mode/2up?view=theater">On the partition of an even number into two primes</a>, Proc. London Math. Soc., ser. I, vol.4 (1871), pp. 4-6 (Math. Papers, vol. 2, pp. 709-711).
%H A380839 Marek Wolf, <a href="https://pracownicy.uksw.edu.pl/mwolf/budap.pdf">Applications of statistical mechanics in number theory</a>, Physica A 274 (1999) 149-157 (formula (9) p. 154).
%F A380839 a(n) = numerator(A173557(n)/A305444(n)).
%F A380839 a(p^n) = p - 1 for prime p.
%F A380839 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A307410(k) = Product_{prime p > 2} (1 + 1/(p*(p-2))) = 1.51478012... (A167864). - _Amiram Eldar_, Mar 03 2025
%e A380839 1, 1, 2, 1, 4/3, 2, 6/5, 1, 2, 4/3, 10/9, 2, 12/11, ...
%e A380839 a(35) = 8 because 35 = 5 * 7 and then product is ((5-1)/(5-2))*((7-1)/(7-2)) = 8/5.
%t A380839 j = {}; Do[prod = 1; Do[If[PrimeQ[n] && IntegerQ[d/n], prod = prod (n - 1)/(n - 2)], {n, 3, d}]; AppendTo[j, prod], {d, 1, 74}]; Numerator[j]
%t A380839 f[p_, e_] := If[p == 2, 1, (p-1)/(p-2)]; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Mar 03 2025 *)
%o A380839 (PARI) a(n) = my(f=factor(n)[,1]); numerator(prod(k=1, #f, if ((p=f[k])>2, (p-1)/(p-2), 1))); \\ _Michel Marcus_, Feb 05 2025
%Y A380839 Cf. A167864, A173557, A305444, A307410 (denominators).
%K A380839 nonn,frac
%O A380839 1,3
%A A380839 _Artur Jasinski_, Feb 05 2025