This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384455 #8 May 30 2025 10:35:14
%S A384455 0,1,2,5,3,4,6,3,4,1,9,1,4,9,6,7,0,1,1,0,3,9,7,0,6,0,7,2,5,7,1,7,7,1,
%T A384455 6,7,4,6,3,2,9,2,5,7,2,2,3,3,3,1,0,5,1,7,2,2,6,5,1,5,2,1,5,7,3,1,6,3,
%U A384455 0,0,7,1,0,5,9,1,8,9,1,8,1,6,1,8,2,9,1,6,4,1,7,2,3,3,8,6,1,7,0,9,2,9,9,0,9,0
%N A384455 Decimal expansion of Sum_{k>=2} (-1)^k*P(k)/(k+1) - M/2 (negated), where P(s) is the prime zeta function and M is Mertens's constant.
%C A384455 The constant C in Theorem 2.2 in Jakimczuk (2025): Product_{p prime <= x} (1 + 1/p)^p = exp(PrimePi(x) + C)/sqrt(log(x)) + O(exp(PrimePi(x))/(sqrt(log(x))*exp(a*sqrt(log(x))))), where a is a positive constant, and Product_{k=1..n} (1 + 1/prime(k))^prime(k) = (exp(n + C)/sqrt(log(n))) * (1 - log(log(n))/(2*log(n)) + o(log(log(n))/log(n))).
%H A384455 Rafael Jakimczuk, <a href="http://dx.doi.org/10.13140/RG.2.2.13911.18084">Two Topics in Number Theory: Products Related with the e Number and Sum of Subscripts in Prime Numbers</a>, ResearchGate, May 2025. See p. 3.
%F A384455 Equals -gamma/2 + Sum_{k>=2} (1/(2*k) + (-1)^k/(k+1)) * P(k), where P(s) is the prime zeta function and gamma is Euler's constant (A001620).
%e A384455 -0.01253463419149670110397060725717716746329257223331...
%o A384455 (PARI) suminf(k = 2, (1/(2*k) + (-1)^k/(k+1)) * sumeulerrat(1/p^k)) - Euler/2
%Y A384455 Cf. A000720 (PrimePi), A001620, A077761 (Mertens's constant), A110544 (analogous with product over positive integers), A229495.
%K A384455 nonn,cons
%O A384455 0,3
%A A384455 _Amiram Eldar_, May 30 2025