A014601 -id:A014601 - cp's OEIS frontend

A106753 Discriminants, negated, of definite binary quadratic forms.

3, 7, 8, 11, 12, 15, 19, 20, 23, 24, 27, 28, 31, 32, 35, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 67, 68, 71, 72, 75, 76, 79, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 103, 104, 107, 108, 111, 112, 115, 116, 119, 120, 123, 124, 127, 128, 131, 132, 135, 136, 139

Offset: 1

Steven Finch, May 16 2005

A definite binary quadratic form a*x^2 + b*x*y + c*y^2 over the integers has discriminant D = b^2 - 4*a*c < 0; -D is assumed to be a nonsquare.

Cf. A014601, A042948 (with squares), A079896 (indefinite case).

Select[ Range[148], (Mod[ -#, 4] == 0 || Mod[ -#, 4] == 1) && !IntegerQ[Sqrt[ # ]] & ]

-a(n) is 0 (mod 4) or 1 (mod 4), but not a square.

A116192 Triangle T(n,k), 0<=k<=n : T(n,k)is smallest number such that T(n,k)>T(n-1,k-1), T(n,k)>T(n-1,k), T(n,k)and T(n-1,k-1)+T(n-1,k) have the same parity, T(0,0)=1 .

Original entry on oeis.org

1, 3, 3, 5, 4, 5, 7, 7, 7, 7, 9, 8, 8, 8, 9, 11, 11, 10, 10, 11, 11, 13, 12, 13, 12, 13, 12, 13, 15, 15, 15, 15, 15, 15, 15, 15, 17, 16, 16, 16, 16, 16, 16, 16, 17, 19, 19, 18, 18, 18, 18, 18, 18, 19, 19, 21, 20, 21, 20, 20, 20, 20, 20, 21, 20, 21, 23, 23, 23, 23, 22, 22, 22, 22

Offset: 0

Philippe Deléham, Apr 08 2007

Sequence read mod 2 gives A047999 .

			Triangle begins:
1;
3, 3;
5, 4, 5;
7, 7, 7, 7;
9, 8, 8, 8, 9;
11, 11, 10, 10, 11, 11;
13, 12, 13, 12, 13, 12, 13;
15, 15, 15, 15, 15, 15, 15, 15 ;...

Diagonals : A005408, A014601.

Sum_{k, 0<=k<=n}T(n,k)=A128806(n). T(2n,n)=A008574(n). T(n,k)=2n+1 if binomial(n,k) is odd, T(n,k)=2n if binomial(n,k) is even .

A139692 Nonnegative discriminants of the monic general quartic polynomials with integer coefficients.

Original entry on oeis.org

0, 5, 12, 20, 21, 32, 40, 45, 48, 49, 60, 77, 81, 85, 96, 104, 112, 117, 125, 140, 144, 148, 165, 169, 189, 192, 216, 221, 224, 229, 252, 256, 257, 260, 272, 285, 288, 320, 321, 333, 357, 361, 392, 400, 404, 432, 437, 468, 469, 473, 480, 488, 500, 512, 525, 528, 533, 544, 549

Offset: 1

Artur Jasinski, Apr 29 2008

nonn

Available discriminants of the general normalized quartic polynomial x^4+c*x^3+d*x^2+e*x+f with c,d,e,f >= 0.

Cf. A014601, A042948, A139691.

aa = {}; b = 1; Do[Print[f]; Do[Do[Do[k = c^2 d^2 e^2 - 4 b d^3 e^2 - 4 c^3 e^3 + 18 b c d e^3 - 27 b^2 e^4 - 4 c^2 d^3 f + 16 b d^4 f + 18 c^3 d e f - 80 b c d^2 e f - 6 b c^2 e^2 f + 144 b^2 d e^2 f - 27 c^4 f^2 + 144 b c^2 d f^2 - 128 b^2 d^2 f^2 - 192 b^2 c e f^2 + 256 b^3 f^3; If[k >= 0 && k < 1000, AppendTo[aa, k]], {c, 0, 30}], {d, 0, 30}], {e, 0, 30}], {f, 0, 30}]; Union[aa]

Edited, and some missing terms inserted, by Robert Israel, May 16 2018

A250203 Numbers n such that the Phi_n(2) is the product of exactly two primes and is divisible by 2n+1.

Original entry on oeis.org

11, 20, 23, 35, 39, 48, 83, 96, 131, 231, 303, 375, 384, 519, 771, 848, 1400, 1983, 2280, 2640, 2715, 3359, 6144, 7736, 7911, 11079, 13224, 16664, 24263, 36168, 130439, 406583

Offset: 1

Eric Chen, Mar 13 2015

Here Phi_n is the n-th cyclotomic polynomial.
Is this sequence infinite?
Phi_n(2)/(2n+1) is only a probable prime for n > 16664.
a(33) > 2000000.
Subsequence of A005097 (2 * a(n) + 1 are all primes)
Subsequence of A081858.
2 * a(n) + 1 are in A115591.
Primes in this sequence are listed in A239638.
A085021(a(n)) = 2.
All a(n) are congruent to 0 or 3 (mod 4). (A014601)
All a(n) are congruent to 0 or 2 (mod 3). (A007494)
Except the term 20, all even numbers in this sequence are divisible by 8.

			Phi_11(2) = 23 * 89 and 23 = 2 * 11 + 1, so 11 is in this sequence.
Phi_35(2) = 71 * 122921 and 71 = 2 * 35 + 1, so 35 is in this sequence.
Phi_48(2) = 97 * 673 and 97 = 2 * 48 + 1, so 48 is in this sequence.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Will Edgington, Factorization of completely factored Phi_n(2) [from Internet Archive Wayback Machine]
Henri Lifchitz and Renaud Lifchitz, PRP records. Search for (2^a-1)/b
Samuel Wagstaff, The Cunningham project

Cf. A239638, A085724, A072226, A005384, A005097, A081858, A014601, A014664, A001917, A115591.

Select[Range[10000], PrimeQ[2*# + 1] && PowerMod[2, #, 2*# + 1] == 1 &&
PrimeQ[Cyclotomic[#, 2]/(2*#+1)] &]

isok(n) = if (((x=polcyclo(n, 2)) % (2*n+1) == 0) && (omega(x) == 2), print1(n, ", ")); \\ Michel Marcus, Mar 13 2015

cp's OEIS Frontend

A106753 Discriminants, negated, of definite binary quadratic forms.

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A116192 Triangle T(n,k), 0<=k<=n : T(n,k)is smallest number such that T(n,k)>T(n-1,k-1), T(n,k)>T(n-1,k), T(n,k)and T(n-1,k-1)+T(n-1,k) have the same parity, T(0,0)=1 .

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A139692 Nonnegative discriminants of the monic general quartic polynomials with integer coefficients.

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A250203 Numbers n such that the Phi_n(2) is the product of exactly two primes and is divisible by 2n+1.

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