A007634 -id:A007634 - cp's OEIS frontend

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.

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

A260678 Numbers n>0 for which n+(17-n)^2 is not prime.

Original entry on oeis.org

33, 34, 37, 42, 49, 50, 51, 53, 56, 58, 60, 65, 67, 68, 69, 71, 72, 75, 78, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97, 100, 101, 102, 105, 106, 107, 110, 111, 113, 114, 116, 117, 118, 119, 122, 123, 124, 128, 129, 132, 133, 134, 135, 136, 139, 141, 143, 148, 151, 152, 153

Offset: 1

M. F. Hasler, Nov 15 2015

Motivated by the fact that n+(17-n)^2 = 1+16^2, 2+15^2, ..., 16+1^2, 17+0^2, 18+1^2, 19+2^2, ..., 32+15^2 are all prime. This has an explanation through Heegener numbers, similar to Euler's prime-generating polynomial, cf. A002837 and related crossrefs.

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

Cf. A260679 (n+(17-n)^2), A007635 (primes in that sequence = primes of the form n^2+n+17).

Cf. A002837 (n^2-n+41 is prime), A005846 (primes of form n^2+n+41), A007634 (n^2+n+41 is composite), A097823 (n^2+n+41 is not squarefree).

[n: n in [1..180] | not IsPrime(n+(17-n)^2)]; // Vincenzo Librandi, Nov 16 2015

remove(t -> isprime(t+(17-t)^2), [$1..200]); # Robert Israel, May 02 2017

Select[Range[200], !PrimeQ[# + (17 - #)^2] &] (* Vincenzo Librandi, Nov 16 2015 *)

for(n=1,999,isprime(n+(17-n)^2)||print1(n","))

A268405 Numbers m such that m^2 + m + 41 is a product of 7 primes.

Original entry on oeis.org

3019035, 3312609, 4005577, 4205871, 4270887, 4502832, 4838229, 4933775, 5086008, 6142338, 6618260, 6932403, 6941996, 7263518, 7375900, 7643466, 7939002, 8268798, 8473961, 8485664, 8499341, 8892530, 8978097, 8991587, 9075462, 9317324, 9469974, 9709914, 9736792, 9745217

Offset: 1

Zak Seidov, Feb 04 2016

nonn

Note that a(1) = 3019035 = A228122(7) and a(11) = 6618260 = A145293(7).

Zak Seidov, Table of n, a(n) for n = 1..140

Cf. A145293, A228122, A268290, A005846, A007634, A145292, A145294, A145295.

A273597 min { x >= 0 | A273595(n) + prime(n)*x + x^2 is composite }, where A273595(n) is such that this is a local maximum.

Original entry on oeis.org

39, 38, 37, 35, 34, 32, 31, 29, 26, 25, 22, 20, 19, 17, 14, 12, 11, 12, 12, 12, 12, 16, 15, 12, 12, 13, 14, 13, 13, 14, 13, 13, 13, 13, 14, 14, 14, 16, 16, 16, 15, 15, 16, 16, 17

Offset: 2

M. F. Hasler, May 26 2016

nonn

See A273595 for further information and (cross)references.
From the initial values, the sequence seems strictly decreasing, with a(n+1) - a(n) = (prime(n+1) - prime(n))/2; however, this property does not persist beyond n = 16.
This is the subsequence of A273770 with indices n corresponding to odd primes 2n+1, see formula. - M. F. Hasler, Feb 17 2020

Index to entries related to primes produced by polynomials.

Cf. A002837, A007634, A005846, A097823, A144051, A187057 .. A187060, A190800, A191456 ff.

A273597(n)=A273770(prime(n)\2) \\ changed Feb 17 2020

A273597(n,p=prime(n),q=A273756(p\2))={for(x=1,oo,isprime(q+p*x+x^2)||return(x))} \\ M. F. Hasler, Feb 17 2020

a(n) = (81 - prime(n))/2 for 1 < n < 17.

a(n) = A273770((prime(n) - 1)/2). - M. F. Hasler, Feb 17 2020

Edited and extended using A273756(0..100) due to Don Reble, by M. F. Hasler, Feb 17 2020

A260679 a(n) = n + (17 - n)^2.

Original entry on oeis.org

257, 227, 199, 173, 149, 127, 107, 89, 73, 59, 47, 37, 29, 23, 19, 17, 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 289, 323, 359, 397, 437, 479, 523, 569, 617, 667, 719, 773, 829, 887, 947, 1009, 1073, 1139, 1207, 1277, 1349, 1423, 1499, 1577, 1657

Offset: 1

M. F. Hasler, Nov 15 2015

Motivated by the fact that the first 32 terms of this sequence are primes. This has an explanation through Heegener numbers, similar to Euler's prime-generating polynomial (cf. A002837 and related crossrefs).
See also A007635 for the primes in this sequence, A260678 for indices k for which a(k) is composite.
Sequence provides all numbers m for which 4*m - 67 is a square. - Bruno Berselli, Nov 16 2015

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A007635 (primes in this sequence = primes of the form n^2 + n + 17).
Cf. A002837 (n^2 - n + 41 is prime), A005846 (primes of form n^2 + n + 41), A007634 (n^2 + n + 41 is composite), A097823 (n^2 + n + 41 is not squarefree).
Cf. A260678.

[n+(17-n)^2: n in [1..70]]; // Vincenzo Librandi, Nov 16 2015

Table[n + (17 - n)^2, {n, 70}] (* Vincenzo Librandi, Nov 16 2015 *)
LinearRecurrence[{3,-3,1},{257,227,199},60] (* Harvey P. Dale, May 12 2019 *)

PARI
```
for(n=1,99,print1(n+(17-n)^2,","))
```

G.f.: x*(257 - 544*x + 289*x^2)/(1 - x)^3.
From Elmo R. Oliveira, Feb 11 2025: (Start)
E.g.f.: exp(x)*(x^2 - 32*x + 289) - 289.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A284043 Starts of a run of at least n consecutive numbers k for which k^2 - k + 41 is composite.

Original entry on oeis.org

41, 41, 122, 162, 299, 326, 326, 1064, 1064, 1064, 1064, 1064, 5664, 5664, 5664, 5664, 9265, 9265, 9265, 22818, 22818, 37784, 37784, 47494, 100202, 100202, 100202, 167628, 167628, 167628, 167628, 167628, 167628, 167628, 167628, 176956, 176956, 176956, 1081297

Offset: 1

Amiram Eldar, Jun 14 2017

nonn

This sequence is inspired by the problem proposed by Sidney Kravitz in 1963: "It is known that f(n)=n^2-n+41 yields prime numbers for n=1, 2, ..., 40. Find a sequence of 40 consecutive values of n for which f(n) is composite." Lawrence A. Ringenberg and others suggested the solution that starts at f(1)*f(2)*...*f(40)+1 (about 4.890... * 10^101). B. A. Hausmann suggested the smaller solution that starts at f(1)*f(2)*...*f(20)-19 (about 3.213... * 10^42). The smallest solution is a(40) = 1081297.

			The values of f(n)=n^2-n+41 at 122, 123 and 124 are: 14803 = 113*131, 15047 = 41*367 and 15293 = 41*373. This is the first case of 3 consecutive composite values, thus a(3) = 122.

Thomas Koshy, Elementary Number Theory with Applications, Academic Press, 2nd edition, 2007, Chapter 2, p. 147, exercise 50.

Amiram Eldar, Table of n, a(n) for n = 1..130 (terms below 10^10)
Sidney Kravitz, Problem 527, Mathematics Magazine, Vol. 36, No. 4 (1963), p. 264.
Lawrence A. Ringenberg et al., A Prime Generator, Solutions to Problem 527, Mathematics Magazine, Vol. 37, No. 2 (1964), pp. 122-123.

Cf. A002837, A005846, A007634, A056561, A145292, A202018.

f[n_] := n^2 - n + 41; a = PrimeQ[f[Range[1, 10^7]]]; b = Split[a]; c = Length /@ b; d = Accumulate[c]; nc = Length[c]; e = {}; For[len = 0, len < 100, len++; k = 2;  While[k <= nc && c[[k]] < len, k += 2]; If[k <= nc && c[[k]] >= len, ind = d[[k - 1]] + 1; e = AppendTo[e, ind]]]; e

cp's OEIS Frontend

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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A260678 Numbers n>0 for which n+(17-n)^2 is not prime.

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A268405 Numbers m such that m^2 + m + 41 is a product of 7 primes.

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A273597 min { x >= 0 | A273595(n) + prime(n)*x + x^2 is composite }, where A273595(n) is such that this is a local maximum.

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A260679 a(n) = n + (17 - n)^2.

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A284043 Starts of a run of at least n consecutive numbers k for which k^2 - k + 41 is composite.

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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 = 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.