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A256546 Numbers n such that n^4 + (n+1)^4 + ... + (n+k)^4 is composite for every k>=0.

%I A256546 #23 Sep 08 2022 08:46:11
%S A256546 11,17,18,22,29,32,35,39,41,44,46,49,50,51,53,55,57,59,60,61,64,66,69,
%T A256546 70,73,75,76,77,79,81,86,92,95,96,101,102,103,107,112,113,114,116,117,
%U A256546 118,120,125,131,133,135,137,138,141,143,144,147,148,149,150,151
%N A256546 Numbers n such that n^4 + (n+1)^4 + ... + (n+k)^4 is composite for every k>=0.
%C A256546 Number n is in the sequence if and only if the following seven numbers are all composite:
%C A256546 P_1(n) = 2n^4 + 4n^3 + 6n^2 + 4n + 1,
%C A256546 P_2(n) = 3n^4 + 12n^3 + 30n^2 + 36n + 17,
%C A256546 P_3(n) = 5n^4 + 40n^3 + 180n^2 + 400n + 354,
%C A256546 P_4(n) = 6n^4 + 60n^3 + 330n^2 + 900n + 979,
%C A256546 P_5(n) = 10n^4 + 180n^3 + 1710n^2 + 8100n + 15333,
%C A256546 P_6(n) = 15n^4 + 420n^3 + 6090n^2 + 44100n + 127687,
%C A256546 P_7(n) = 30n^4 + 1740n^3 + 51330n^2 + 756900n + 4463999.
%C A256546 For a generalization, see comment in A256581.
%H A256546 Peter J. C. Moses, <a href="/A256546/b256546.txt">Table of n, a(n) for n = 1..1000</a>
%o A256546 (Magma) [n: n in [0..2*10^2] | not IsPrime(2*n^4+4*n^3+6*n^2 +4*n+1) and not IsPrime(3*n^4+12*n^3+30*n^2+36*n+17) and not IsPrime(5*n^4+40*n^3+180*n^2+400*n+354) and not IsPrime(6*n^4+60*n^3+330*n^2+900*n+979) and not IsPrime(10*n^4+ 180*n^3+1710*n^2+8100*n+15333) and not IsPrime(15*n^4+ 420*n^3+6090*n^2+44100*n+127687) and not IsPrime(30*n^4+ 1740*n^3+51330*n^2+756900*n+4463999)]; // _Vincenzo Librandi_, Apr 03 2015
%Y A256546 Cf. A000583, A256385, A256503, A089306, A077654, A256385, A256581.
%K A256546 nonn
%O A256546 1,1
%A A256546 _Vladimir Shevelev_ and _Peter J. C. Moses_, Apr 01 2015