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A005098 Numbers k such that 4k + 1 is prime.

%I A005098 #74 Feb 16 2025 08:32:28
%S A005098 1,3,4,7,9,10,13,15,18,22,24,25,27,28,34,37,39,43,45,48,49,57,58,60,
%T A005098 64,67,69,70,73,78,79,84,87,88,93,97,99,100,102,105,108,112,114,115,
%U A005098 127,130,135,139,142,144,148,150,153,154,160,163,165,168,169,175,177,183
%N A005098 Numbers k such that 4k + 1 is prime.
%C A005098 Sum of i-th and j-th triangular numbers, where i=A096029(n), j=A096030(n); i.e., a(n) = A000217(A096029(n)) + A000217(A096030(n)). - _Lekraj Beedassy_, Jun 16 2004
%C A005098 For every k in the sequence, there is exactly 1 square number that can be subtracted to leave a pronic (A002378). E.g., 27 - 25 = 2, 99 - 9 = 90. - _Jon Perry_, Nov 06 2010
%C A005098 See A208295 for details concerning the preceding _Jon Perry_ comment. - _Wolfdieter Lang_, Mar 29 2012
%C A005098 a(k) appears in the o.g.f. for floor(A002144(k)*j^2/4), j >= 0, for k >= 1: x*(a(k)*(1 + x^2) + b(k)*x)/((1 - x)^3*(1 + x)), together with b(k) = (A002144(k) + 1)/2 = A119681(k). - _Wolfdieter Lang_, Aug 07 2013
%H A005098 Zak Seidov, <a href="/A005098/b005098.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A005098 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WilsonsTheorem.html">Wilson's Theorem</a>.
%F A005098 a(n) = (A002144(n)-1)/4.
%p A005098 a := []; for k from 1 to 500 do if isprime(4*k+1) then a := [op(a), k]; fi; od: A005098 := k->a[k];
%t A005098 Select[Range[200], PrimeQ[4# + 1] &] (* _Harvey P. Dale_, Apr 20 2011 *)
%o A005098 (Magma) [k: k in [0..10000] | IsPrime(4*k+1)] // _Vincenzo Librandi_, Nov 18 2010
%o A005098 (PARI) is(k)=isprime(4*k+1) \\ _Charles R Greathouse IV_, Nov 20 2012
%o A005098 (Haskell)
%o A005098 a005098 = (`div` 4) . (subtract 1) . a002144
%o A005098 -- _Reinhard Zumkeller_, Mar 17 2013
%Y A005098 See A002144 for the actual primes.
%Y A005098 Cf. A002972, A002973, A173331, A173330, A023212.
%K A005098 nonn,nice,easy
%O A005098 1,2
%A A005098 _N. J. A. Sloane_
%E A005098 More terms from _Ray Chandler_, Jun 26 2004
%E A005098 Edited by _Charles R Greathouse IV_, Mar 17 2010