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A082542 a(n) = prime(n) + 2 - (prime(n) mod 4).

%I A082542 #39 Aug 24 2025 12:49:12
%S A082542 2,2,6,6,10,14,18,18,22,30,30,38,42,42,46,54,58,62,66,70,74,78,82,90,
%T A082542 98,102,102,106,110,114,126,130,138,138,150,150,158,162,166,174,178,
%U A082542 182,190,194,198,198,210,222,226,230,234,238,242,250,258,262,270,270,278
%N A082542 a(n) = prime(n) + 2 - (prime(n) mod 4).
%C A082542 For k > 1: a(k+1) = a(k) if and only if prime(k) == 1 modulo 4 and prime(k+1) = prime(k) + 2, see A071695 and A071696.
%H A082542 G. C. Greubel, <a href="/A082542/b082542.txt">Table of n, a(n) for n = 1..10000</a>
%F A082542 a(n) = A000040(n) + A070750(n).
%F A082542 a(n+1) = p + (-1/p) = p + (-1)^((p-1)/2), where p is the n-th odd prime and (-1/p) denotes the value of Legendre symbol. - _Lekraj Beedassy_, Mar 17 2005
%F A082542 a(n) = (A000040(n) OR 3) - 1. - _Jon Maiga_, Nov 14 2018
%F A082542 From _Amiram Eldar_, Dec 24 2022: (Start)
%F A082542 a(n) = A100484(n) - A076342(n).
%F A082542 Product_{n>=1} a(n)/prime(n) = 2/Pi (A060294). (End)
%e A082542 a(2) = 2 because the second prime is 3, and 3 + 2 - 3 = 2.
%e A082542 a(3) = 6 because the third prime is 5, and 5 + 2 - 1 = 6.
%e A082542 a(4) = 6 because the fourth prime is 7, and 7 + 2 - 3 = 6.
%t A082542 Table[Prime[n] + 2 - Mod[Prime[n], 4], {n, 60}] (* _Alonso del Arte_, Feb 23 2015 *)
%t A082542 #+2-Mod[#,4]&/@Prime[Range[60]] (* _Harvey P. Dale_, Aug 24 2025 *)
%o A082542 (PARI) vector(60, n, 2 + prime(n) - lift(Mod(prime(n),4))) \\ _G. C. Greubel_, Nov 14 2018
%o A082542 (Magma) [2 + NthPrime(n) - (NthPrime(n) mod 4): n in [1..60]]; // _G. C. Greubel_, Nov 14 2018
%Y A082542 Cf. A000040, A060294, A070750, A076342.
%Y A082542 Cf. A039702, A071695, A071696, A100484.
%K A082542 nonn,easy,changed
%O A082542 1,1
%A A082542 _Reinhard Zumkeller_, May 02 2003