login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

CROSSREFS

Cf. A007634, A055390, A201998, and with division, A235381.

Sequence in context: A184602 A144547 A259403 * A185502 A035895 A035775

Adjacent sequences: A241526 A241527 A241528 * A241530 A241531 A241532

KEYWORD

nonn

AUTHOR

Matt C. Anderson, Apr 27 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 8 09:22 EST 2023. Contains 360138 sequences. (Running on oeis4.)