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

11, 17, 18, 22, 29, 32, 35, 39, 41, 44, 46, 49, 50, 51, 53, 55, 57, 59, 60, 61, 64, 66, 69, 70, 73, 75, 76, 77, 79, 81, 86, 92, 95, 96, 101, 102, 103, 107, 112, 113, 114, 116, 117, 118, 120, 125, 131, 133, 135, 137, 138, 141, 143, 144, 147, 148, 149, 150, 151

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, Apr 01 2015

Keywords

• nonn

Comments

Number n is in the sequence if and only if the following seven numbers are all composite:

P_1(n) = 2n^4 + 4n^3 + 6n^2 + 4n + 1,

P_2(n) = 3n^4 + 12n^3 + 30n^2 + 36n + 17,

P_3(n) = 5n^4 + 40n^3 + 180n^2 + 400n + 354,

P_4(n) = 6n^4 + 60n^3 + 330n^2 + 900n + 979,

P_5(n) = 10n^4 + 180n^3 + 1710n^2 + 8100n + 15333,

P_6(n) = 15n^4 + 420n^3 + 6090n^2 + 44100n + 127687,

P_7(n) = 30n^4 + 1740n^3 + 51330n^2 + 756900n + 4463999.

For a generalization, see comment in A256581.

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000583, A256385, A256503, A089306, A077654, A256385, A256581.

Programs

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