A241529 - OEIS

A241529

Positive numbers k such that k^2 + k + 41 is composite and there are no positive integers a,c,d such that k = c*a*z^2 + ((((d+2)*(1/3))*c-2)*a/d+1)*z + ((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2 - (((d-1)*(1/3))*c+1)/d)/c for an integer z.

2887, 2969, 3056, 3220, 3365, 3464, 3565, 3611, 3719, 3746, 3814, 3836, 3874, 3879, 3955, 4142, 4147, 4211, 4277, 4371, 4403, 4483, 4564, 4572, 4661, 4730, 4813, 4881, 4888, 4902, 4906, 4965, 4982, 5132, 5175, 5208, 5410, 5431, 5509, 5527, 5564, 5624, 5669

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OFFSET

1,1

COMMENTS

This sequence has a restriction involving 4 variables. More composite cases are described with a better restrictive expression. The expression for k(a,c,d,z) will force k^2 + k + 41 to be either a fraction or a composite number.

The condition on k(a,c,d,z) was determined by quadratic curve fitting. It has been automated with the Maple Interactive() command. The ultimate motivation is to try to find a closed-from expression that generates all the composite cases of k^2 + k + 41 for integer k.

What is the smallest value of n where this sequence's a(n) < 2n? (For A194634, this value is 2358.) - J. Lowell, Feb 25 2019

REFERENCES

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

LINKS

Table of n, a(n) for n=1..43.

Matt C. Anderson, Graph of composite values for n^2 + n + 41 with a modular symmetry.

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

MAPLE

# Euler considered the prime values for n^2 + n + 41;

# This is a 76 second calculation on a 2.93 GHz machine

h := n^2+n+41;

y := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c;

y2 := subs(n = y, h);

y3 := factor(y2);

# note that y is an expression in 4 variables.

# After a composition of functions, an algebraic factorization

# can be observed in y3. As long as y3 is an integer, it will

# be composite. This is because y3 factors and both factors

# are integers bigger than one.

maxn := 6000;

A := {}:

for n to maxn do

g := n^2+n+41:

if isprime(g) = false then A := `union`(A, {n}) end if :

end do:

# now the A set contains composite values of the form

# n^2 + n + 41 less than maxn.

c := 1: a := 1: d := 1: z := -1: p := 41:

q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c:

A2 := A:

while q < maxn do

while `and`(q < maxn, d < 100) do

while q < maxn do while

q < maxn do

A2 := `minus`(A2, {q});

A2 := `minus`(A2, {c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c});

z := z+1;

A2 := `minus`(A2, {c*a*z^2-((((d+2)*(1/3))*c-2)*a/d+1)*(1*z)+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c}); q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c

end do;

a := a+1; z := -1;

q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c :

end do;

d := d+1: a := 1:

q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c :

end do:

c := c+1: d := 1:

q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c :

end do:

A2;

# Matt C. Anderson, May 13 2014