A055390 -id:A055390 - 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

A194634 Numbers n such that k= n^2 + n + 41 is composite and there is no integer x such that n= x^2 + 40; n= (x^2+x)/2 + 81; or n= 3*x^2 - 2x + 122.

Original entry on oeis.org

127, 138, 155, 163, 164, 170, 173, 178, 185, 190, 204, 205, 207, 208, 213, 215, 216, 232, 237, 239, 242, 244, 245, 246, 248, 249, 251, 256, 259, 261, 266, 268, 270, 278, 279, 283, 284, 286, 287, 289, 295, 299, 300, 301, 302, 309, 314, 321, 325, 326, 327, 328

Offset: 1

Matt C. Anderson, Aug 30 2011

The parabola curve fit: p1(0)=40; p1(1)=41; p1(2)=44 yields p1(x)=x^2+40. A second fit: p2(0)=81; p2(1)=82; p2(2)=84 yields p2(x)=(x^2+x)/2 + 81. A third fit: p3(0)=122; p3(1)=123; p3(2)=130 yields p3(x)=3x^2-2*x+122.
Substituting n=x^2 into k=n^2+n+41 is factorable as: k1=(x^2+x+41)*(x^2-x+41).  This shows that all lattice points on p1 produce a composite k.
Similarly, substituting n=(x^2-x)/2 + 81 into k factors as k2=(x^2+163)*(x^2+2*x+164)/4. So all lattice points on p2 produce a composite k.
Similarly, substituting n=3*x^2-2*x+122 into k factors as k3=(x^2-x+41)*(9*x^2-3*x+367).  So all lattice points on p3 produce a composite k.
This procedure can be continued with p4(x)=3*x^2+8*x+127, p5(x)=4*x^2-3*x+163, p6(x)=4*x^2+11*x+170, p7(x)=5*x^2-4*x+204, p8(x)=5*x^2+14*x+213, p9(x)=(3*x^2-x)/2+244, p10(x)=(3*x^2+7*x)/2+246, and so on.

John Stillwell, Elements of Number Theory, Springer, 2003, page 3.
R. Crandall and C. Pomerance, Prime Numbers A Computational Perspective 2nd ed., Springer, 2005, page 21.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A007634 (n such that n^2+n+41 is composite).
Cf. A055390 (members of A007634 that are not lattice points of x^2+40).
Cf. A194565 (members of A055390 that are not lattice points of (x^2+x)/2 + 81).

A007634:={}:
for n from 1 to 1000 do
k:=n^2+n+41:
if isprime(k)=false then
A007634:=A007634 union {n}:
end if:
end do:
pv1:=Vector(1000,j->(j-1)^2+40):
p1:=convert(pv1,set):
A055390:=A007634 minus p1 minus {0}:
pv2:=Vector(1000,j->((j-1)^2+(j-1))/2+81):
p2:=convert(pv2,set):
A194565:=A055390 minus p2:
pv3:=Vector(1000,j->(3*(j-1)^2-2*(j-1)+122)):
p3:=convert(pv3,set):
p3set:=A194565 minus p3;

is(n)=!isprime(n^2+n+41) && !issquare(n-40) && !issquare(8*n-647) && n > 126 && (x->3*x^2-2*x+122)(round((1+sqrt(3*n-365))/3))!=n \\ Charles R Greathouse IV, Apr 25 2014

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

Fixed subscripts in first comment. Added * in 4th comment. Added 5th comment. Changed g to k for consistancy. Improved Maple code. Added second book reference. Changed name to agree with comment of editor.

A194565 Numbers n such that n^2 + n + 41 is composite and n is not a lattice point on the parabolas p1 = x^2 + 40 or p2 = (x^2+x)/2 + 81.

Original entry on oeis.org

122, 123, 127, 130, 138, 143, 155, 162, 163, 164, 170, 173, 178, 185, 187, 190, 204, 205, 207, 208, 213, 215, 216, 218, 232, 237, 239, 242, 244, 245, 246, 248, 249, 251, 255, 256, 259, 261, 266, 268, 270, 278, 279, 283, 284, 286, 287, 289, 295, 298, 299, 300

Offset: 1

Matt C. Anderson, Aug 28 2011

The parabola curve fit: p1(0)=40; p1(1)=41; p1(2)=44 yields p1(x)=x^2+40.  A second fit: p2(0)=81; p2(1)=82; p2(2)=84 yields p2(x)=(x^2+x)/2 + 81.
Substituting n=x^2+40 into g=n^2+n+41 is factorable as: g1=(x^2+x+41)*(x^2-x+41).  This shows that all lattice points on p1 produce a composite g.
Similarly, substituting n=(x^2-x)/2 + 81 into g factors as g2=(x^2+163)*(x^2+2*x+164)/4. So all lattice points on p2 produce a composite g.

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric W. Weisstein, MathWorld: Prime-Generating Polynomial.

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

Cf. A055390 (members of A007634 that are not lattice points of x^2+40).

AV:=Vector(1000,0):
counter:=1:
for n from 1 to 1000 do
g:=n^2+n+41:
if isprime(g)=false then
  AV[counter]:=n:
  counter:=counter+1:
end if
end do:
A007634:=convert(AV,set):
pv1:=Vector(1000,j->(j-1)^2+40):
p1:=convert(pv1,set):
A055390:=A007634 minus p1:
pv2:=Vector(1000,j->((j-1)^2+(j-1))/2+81):
p2:=convert(pv2,set):
ThisSet:=A055390 minus p2 minus {0};

is(n)=!isprime(n^2+n+41) && !issquare(n-40) && !issquare(8*n-647) \\ Charles R Greathouse IV, Apr 25 2014

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

Fixed subscript in first comment by Matt C. Anderson

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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A194634 Numbers n such that k= n^2 + n + 41 is composite and there is no integer x such that n= x^2 + 40; n= (x^2+x)/2 + 81; or n= 3*x^2 - 2x + 122.

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A194565 Numbers n such that n^2 + n + 41 is composite and n is not a lattice point on the parabolas p1 = x^2 + 40 or p2 = (x^2+x)/2 + 81.

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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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Author

Keywords

Comments

References

Links

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Programs

Maple

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.