

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))*c2)*a/d+1)*z + ((((367*d^2+d+1)*(1/9))*c^2((d+2)*(1/3))*c+1)*a/d^2  (((d1)*(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
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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 closedfrom 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, PrimeGenerating 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))*c2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2((d+2)*(1/3))*c+1)*a/d^2(((d1)*(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))*c2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2((d+2)*(1/3))*c+1)*a/d^2(((d1)*(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))*c2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2((d+2)*(1/3))*c+1)*a/d^2(((d1)*(1/3))*c+1)/d)/c});
z := z+1;
A2 := `minus`(A2, {c*a*z^2((((d+2)*(1/3))*c2)*a/d+1)*(1*z)+((((367*d^2+d+1)*(1/9))*c^2((d+2)*(1/3))*c+1)*a/d^2(((d1)*(1/3))*c+1)/d)/c}); q := c*a*z^2+((((d+2)*(1/3))*c2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2((d+2)*(1/3))*c+1)*a/d^2(((d1)*(1/3))*c+1)/d)/c
end do;
a := a+1; z := 1;
q := c*a*z^2+((((d+2)*(1/3))*c2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2((d+2)*(1/3))*c+1)*a/d^2(((d1)*(1/3))*c+1)/d)/c :
end do;
d := d+1: a := 1:
q := c*a*z^2+((((d+2)*(1/3))*c2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2((d+2)*(1/3))*c+1)*a/d^2(((d1)*(1/3))*c+1)/d)/c :
end do:
c := c+1: d := 1:
q := c*a*z^2+((((d+2)*(1/3))*c2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2((d+2)*(1/3))*c+1)*a/d^2(((d1)*(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



