A201998 -id:A201998 - cp's OEIS frontend

A007634 Numbers k such that k^2 + k + 41 is composite.

40, 41, 44, 49, 56, 65, 76, 81, 82, 84, 87, 89, 91, 96, 102, 104, 109, 117, 121, 122, 123, 126, 127, 130, 136, 138, 140, 143, 147, 155, 159, 161, 162, 163, 164, 170, 172, 173, 178, 184, 185, 186, 187, 190, 201, 204, 205, 207, 208, 209, 213, 215, 216, 217

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

A subset of this sequence is shown in A055390. - Matt C. Anderson, Jan 05 2014
If prime p divides m^2+m+41, then m+p*j is in the sequence for all j >= 1. - Robert Israel, Nov 24 2017
Euler noted that the first 40 values of this polynomial, starting with k=0, are all primes. - Harvey P. Dale, Jul 25 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A002837, A055390, A201998.

[n: n in [0..220] | not IsPrime(n^2 + n + 41)]; // Vincenzo Librandi, Sep 28 2012

remove(n -> isprime(n^2+n+41), [$1..1000]); # Robert Israel, Nov 24 2017

Select[Range[220], !PrimeQ[#^2 + # + 41] &] (* Vincenzo Librandi, Sep 28 2012 *)

is(n)=!isprime(n^2+n+41) \\ Charles R Greathouse IV, Apr 25 2014

a(n) ~ n. - Charles R Greathouse IV, Apr 25 2014

A055390 Terms of A007634 where n - 40 is not a square.

Original entry on oeis.org

81, 82, 84, 87, 91, 96, 102, 109, 117, 122, 123, 126, 127, 130, 136, 138, 143, 147, 155, 159, 162, 163, 164, 170, 172, 173, 178, 185, 186, 187, 190, 201, 204, 205, 207, 208, 213, 215, 216, 217, 218, 232, 234, 237, 239, 242, 244, 245, 246, 248, 249, 251, 252

Offset: 1

J. Lowell, Oct 08 2000

nonn

Numbers n such that n^2 + n + 41 is composite and n - 40 is not a square. - Charles R Greathouse IV, Sep 14 2014

Note that if h(n) = n^2 + n + 41, and k(x) = x^2 + 40, then the composition of functions h(k(x)) has an algebraic factorization: h(k(x)) = (x^2 + 40)^2 + (x^2 + 40) + 41 = (x^2 + x + 41)*(x^2 - x + 41). Since both of the expressions in the above product are integers greater than 1, h(k(x)) is composite. - Matt C. Anderson, Oct 24 2012

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007634 (numbers where n^2 + n + 41 is composite). A194634 (numbers in A007634 that are not in 3 parabolas). - Matt C. Anderson, Sep 26 2011

Cf. A201998.

A007634 := {}:
for n from 0 to 1000 do
k := n^2+n+41:
if isprime(k) = false then A007634 := union(A007634, {n}) end if:
end do:
pv1 := Vector(1000, j -> (j-1)^2+40):
p1 := convert(pv1, set):
A055390 := minus(A007634, p1); # Matt C. Anderson, Sep 26 2011
remove(t -> issqr(t-40) or isprime(t^2+t+41), [$1..1000]); # Robert Israel, Nov 24 2017

Select[Range[260],CompositeQ[#^2+#+41]&&!IntegerQ[Sqrt[#-40]]&] (* Harvey P. Dale, Oct 20 2015 *)

is(n)=!isprime(n^2+n+41) && !issquare(n-40) \\ Charles R Greathouse IV, Sep 14 2014

a(n) ~ n. - Charles R Greathouse IV, Sep 14 2014

More terms from David Wasserman, Mar 19 2002

A235381 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c or d such that n = cdx^2 + ((d-2)c + 1)x + ((41d^2 - d + 1)c -1)/d for an integer x.

Original entry on oeis.org

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

Offset: 1

Matt C. Anderson, Jan 08 2014

nonn

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.

			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.

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

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

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

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

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

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

A241529 Positive numbers k such that k^2 + k + 41 is composite and there are no positive integers a,c,d such that k = caz^2 + ((((d+2)(1/3))c-2)a/d+1)z + ((((367d^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.

Original entry on oeis.org

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

Offset: 1

Matt C. Anderson, Apr 27 2014

nonn

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

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

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

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

# 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

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A007634 Numbers k such that k^2 + k + 41 is composite.

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A055390 Terms of A007634 where n - 40 is not a square.

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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.

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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.

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Programs

Maple

A235381 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c or d such that n = cdx^2 + ((d-2)c + 1)x + ((41d^2 - d + 1)c -1)/d for an integer x.

A241529 Positive numbers k such that k^2 + k + 41 is composite and there are no positive integers a,c,d such that k = caz^2 + ((((d+2)(1/3))c-2)a/d+1)z + ((((367d^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.