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A235381 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c or d such that n = c*d*x^2 + ((d-2)*c + 1)*x + ((41*d^2 - d + 1)*c -1)/d for an integer x. 3
611, 622, 630, 663, 679, 734, 758, 835, 867, 966, 978, 995, 1006, 1009, 1060, 1088, 1127, 1142, 1157, 1173, 1175, 1183, 1228, 1280, 1345, 1355, 1368, 1388, 1390, 1426, 1433, 1455, 1457, 1467, 1497, 1538, 1539, 1543, 1554, 1578, 1603, 1612, 1613, 1630, 1661 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Restricting c and d so that c is congruent to 1 modulo d, we have that the composition of functions k(x) factors. k(x) = (1/d^2)*((1 + x*d^2 + x^2*d^2 - d - 2*x*d + 41*d^2)*(c^2*d^2*x^2 + x*d^2*c^2 + 41*c^2*d^2 + 2*x*d*c^2 - 2*x*d*c^2 + c*d - c^2*d + 1). So k(x) is the product of two integers greater than one and is thus composite.

REFERENCES

John Stillwell, Elements of Number Theory, Springer 2003, page 3.

LINKS

Matt C. Anderson, Table of n, a(n) for n = 1..75

EXAMPLE

If d = 1 then n = c*n^2 + (1 - c)*x + 41*c  - 1. This is, up to a change of variables, equivalent to A201998.

MAPLE

maxn := 1000;

A := {};

for n to maxn do

g := n^2+n+41;

if isprime(g) = false then

A := `union`(A, {n}) :

end if :

end do :

A:

# the A list now contains Positive numbers n such that

# n^2 + n + 41 is composite.

# an upper limit for the number of iterations in the

# triple nested while loops is 1000^3 or a billion.

c:=1:

d:=1:

x:=-1:

p:=41:

q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d;

A2:=A:

while q < maxn do

while q < maxn do

while q < maxn do

  A2:=A2 minus {q}:

  A2:=A2 minus {c*x^(2)+(c+1)*x+c*p}:

  A2:=A2 minus {c*d*x^2-((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d}:

  x:=x+1:

  q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:

end do:

c:=c+1:

x:=-1:

q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:

end do:

d:=d+1:

c:=1:

x:=-1:

q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:

end do:

A2

CROSSREFS

Cf. A007634 (numbers n such that n^2 + n + 41 is composite).

Cf. A201998 and A241529 (similar subsequences of A007634).

Sequence in context: A323281 A214855 A072323 * A202373 A232119 A232417

Adjacent sequences:  A235378 A235379 A235380 * A235382 A235383 A235384

KEYWORD

nonn

AUTHOR

Matt C. Anderson, Jan 08 2014

EXTENSIONS

Corrected and edited by Matt C. Anderson, Jan 23 2014

STATUS

approved

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Last modified April 22 06:14 EDT 2019. Contains 322329 sequences. (Running on oeis4.)