A066436 -id:A066436 - cp's OEIS frontend

A331265 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31^2)^2 = y^2.

0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299, 19220, 27683, 39780, 55719, 79856, 114359, 163680, 234183, 327080, 467759, 668856, 956319, 1367240, 1908683, 2728620, 3900699, 5576156, 7971179, 11126940, 15905883, 22737260, 32502539, 46461756, 64854879, 92708600, 132524783

Offset: 1

Mohamed Bouhamida, Feb 12 2020

For the generic case x^2 + (x + p^2)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436, m >= 4 (means p >= 31), the first five consecutive solutions are (0, p^2), (4*m^3+2*m^2-2*m-1, 4*m^4+4*m^3-2*m-1), (8*m^3+8*m^2+4*m, 4*m^4+8*m^3+12*m^2+4*m+1), (12*m^4-40*m^3+44*m^2-20*m+3, 20*m^4-56*m^3+60*m^2-28*m+5), (12*m^4-20*m^3+2*m^2+10*m-4, 20*m^4-28*m^3+14*m-5) and the other solutions are defined by (X(n), Y(n)) = (3*X(n-5) + 2*Y(n-5) + p^2, 4*X(n-5) + 3*Y(n-5) + 2*p^2).

X(n) = 6*X(n-5) - X(n-10) + 2*p^2, and Y(n) = 6*Y(n-5) - Y(n-10) (can be easily proved using X(n) = 3*X(n-5) + 2*Y(n-5) + p^2, and Y(n) = 4*X(n-5) + 3*Y(n-5) + 2*p^2).

			For p=31 (m=4) the first five (5) consecutive solutions are (0, 961), (279, 1271), (656, 1745), (1139, 2389), (1860, 3379).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,6,-6,0,0,0,-1,1).

Cf. A066436 (Primes of the form 2*m^2 - 1).

Solutions x to x^2+(x+p^2)^2=y^2: A118554 (p=7), A207059 (p=17), A309998 (p=23), this sequence (p=31), A332000 (p=47).

I:=[0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299]; [n le 10 select I[n] else 6*Self(n-5) - Self(n-10)+1922: n in [1..100]];

LinearRecurrence[{1, 0, 0, 0, 6, -6, 0, 0, 0, -1, 1}, {0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299, 19220}, 36] (* Jean-François Alcover, Feb 12 2020 *)

concat(0, Vec(x^2*(279 + 377*x + 483*x^2 + 721*x^3 + 1023*x^4 - 217*x^5 - 183*x^6 - 161*x^7 - 183*x^8 - 217*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)) + O(x^30))) \\ Colin Barker, Feb 12 2020

a(n) = 6*a(n-5) - a(n-10) + 1922 for n >= 11; a(1)=0, a(2)=279, a(3)=656, a(4)=1139, a(5)=1860, a(6)=2883, a(7)=4340, a(8)=6419, a(9)=9156, a(10)=13299.
From Colin Barker, Feb 12 2020: (Start)
G.f.: x^2*(279 + 377*x + 483*x^2 + 721*x^3 + 1023*x^4 - 217*x^5 - 183*x^6 - 161*x^7 - 183*x^8 - 217*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)).
a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - a(n-10) + a(n-11) for n>11.
(End)

A124142 Abundant numbers k such that sigma(k) is a perfect power.

Original entry on oeis.org

66, 70, 102, 210, 282, 364, 400, 510, 642, 690, 714, 770, 820, 930, 966, 1080, 1092, 1146, 1164, 1200, 1416, 1566, 1624, 1672, 1782, 2130, 2226, 2250, 2346, 2460, 2530, 2586, 2652, 2860, 2910, 2912, 3012, 3198, 3210, 3340, 3498, 3522, 3560, 3710, 3810

Offset: 1

Walter Kehowski, Dec 01 2006

nonn

Positive integers k such that sigma(k) > 2*k and sigma(k) = a^b where both a and b are greater than 1.

If k is a term with sigma(k) a square, and p and q are members of A066436 that do not divide k, then k*p*q is in the sequence. Thus if A066436 is infinite, so is this sequence. - Robert Israel, Oct 29 2018

			a(1) = 66 since sigma(66) = 144 = 12^2.

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

Cf. A001597, A005101, A065496, A066436.

with(numtheory); egcd := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi; end; L:=[]: for w to 1 do for n from 1 to 10000 do s:=sigma(n); if s>2*n and egcd(s)>1 then print(n,s,ifactor(s)); L:=[op(L),n]; fi od od;

filterQ[n_] := With[{s = DivisorSigma[1, n]}, s > 2n && GCD @@ FactorInteger[s][[All, 2]] > 1];
Select[Range[4000], filterQ] (* Jean-François Alcover, Sep 16 2020 *)

is(k) = {my(s = sigma(k)); s > 2*k && ispower(s);} \\ Amiram Eldar, Aug 02 2024

A143834 Numbers k such that 2k^2 - 1 is not prime.

Original entry on oeis.org

1, 5, 9, 12, 14, 16, 19, 20, 23, 26, 27, 29, 30, 31, 32, 33, 35, 37, 40, 44, 47, 48, 51, 53, 54, 55, 57, 58, 60, 61, 65, 66, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 82, 83, 84, 86, 88, 89, 90, 93, 94, 96, 97, 99, 100, 101, 103, 104, 105, 106, 107, 110, 111, 114, 116, 117

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

Complement of A066049.

Cf. A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830, A143831, A143833.

[n: n in [1..120]| not IsPrime(2*n^2-1)] // Vincenzo Librandi, Jan 28 2011

p = 2; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k],NULL, AppendTo[a, x]], {x, 1, 1000}]; a
Select[Range[120],!PrimeQ[2#^2-1]&] (* Harvey P. Dale, Mar 14 2018 *)

A219280 Smallest prime of the form ChebyshevT[2^n, x].

Original entry on oeis.org

2, 7, 97, 665857, 708158977, 150038171394905030432003281854339710977

Offset: 0

Michel Lagneau, Nov 17 2012

nonn

ChebyshevT[2^n, x] is the 2^n th Chebyshev polynomial of the first kind evaluated at x.
The corresponding numbers x are {2, 2, 2, 3, 2, 8, 164, 29, ...}.
a(7) = T(128, 29) = 2518958009…2561281 contains 226 decimal digits.

			T(1, x) = x => a(0) = T(1,2) = 2 ;
T(2, x) = 2x^2 - 1 => a(1) = T(2, 2) = 7 ;
T(4, x) = 8x^4 - 8x^2 + 1 => a(2) = T(4,2) = 97.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
C. W. Jones, J. C. P. Miller, J. F. C. Conn and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946), 187-203.

Cf. A066436, A144131, A144132, A219276, A219277, A219278, A219279.

Table[k = 0; While[!PrimeQ[ChebyshevT[2^n,k]], k++]; ChebyshevT[2^n,k], {n, 0, 7}]

A242930 Primes of the form (k^2+7)/11.

Original entry on oeis.org

37, 53, 193, 373, 421, 673, 1061, 2213, 2753, 3637, 4481, 5237, 5413, 7333, 7541, 8513, 8737, 9781, 11393, 12853, 14401, 15733, 17761, 19237, 21121, 25153, 25537, 27701, 29537, 34273, 34721, 39841, 42533, 47653, 50593, 51137

Offset: 1

Chai Wah Wu, Jul 10 2014

nonn

Also equal to primes p such that 11*p-7 is a perfect square.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A154616, A002327, A066436.

import sympy
[(k**2+7)/11 for k in range(10**6) if sympy.ntheory.isprime((k**2+7)/11) & ((k**2+7)/11).is_integer()]

A256917 Primes which are not the sums of two consecutive nonsquares.

Original entry on oeis.org

2, 3, 7, 17, 19, 31, 71, 73, 97, 127, 163, 199, 241, 337, 449, 577, 647, 881, 883, 967, 1151, 1153, 1249, 1459, 1567, 1801, 2179, 2311, 2591, 2593, 2887, 3041, 3361, 3527, 3529, 3697, 4049, 4051, 4231, 4801, 4999, 5407, 6271, 6961, 7687, 7937, 8191, 8713, 9521, 10369, 10657

Offset: 1

Juri-Stepan Gerasimov, Apr 23 2015

The union of 2 and A066436 and A090698.

The sums of two consecutive nonsquares are 5, 8, 11, 13, 15, 18, 21, 23, 25, 27, 29, 32, 35, 37, ...

			2, 3, 7 are in this sequence because first three sums of two consecutive nonsquares are 5, 8, 11 and 2, 3, 7 are primes.

Colin Barker and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 800 terms from Barker)

Cf. A000037, A066436, A090698, A154670.

Union[{2},Select[Table[2n^2-1,{n,0,1000}],PrimeQ],Select[Table[2n^2+1,{n,0,1000}],PrimeQ]] (* Ivan N. Ianakiev, Apr 24 2015 *)
Module[{nn=11000,ns},ns=Total/@Partition[Select[Range[nn],!IntegerQ[Sqrt[#]]&],2,1]; Complement[ Prime[Range[PrimePi[Last[ns]]]],ns]] (* Harvey P. Dale, Mar 06 2024 *)

a256917(maxp) = {
  ps=[2];
  k=1; while((t=2*k^2-1)<=maxp, k++; if(isprime(t), ps=setunion(ps, [t])));
  k=1; while((t=2*k^2+1)<=maxp, k++; if(isprime(t), ps=setunion(ps, [t])));
  ps
}
a256917(11000) \\ Colin Barker, Apr 23 2015

list(lim)=my(v=List([2]),t); for(k=2,sqrtint((lim+1)\2), if(isprime(t=2*k^2-1), listput(v,t))); for(k=1,sqrtint((lim-1)\2), if(isprime(t=2*k^2+1), listput(v,t))); Set(v) \\ Charles R Greathouse IV, Apr 23 2015

A219281 Smallest number k such that ChebyshevT[2^n, k] is prime.

Original entry on oeis.org

2, 2, 2, 3, 2, 8, 164, 29, 60, 213, 181, 652

Offset: 0

Michel Lagneau, Nov 17 2012

ChebyshevT[2^n,x] is the 2^n th Chebyshev polynomial of the first kind evaluated at x.

			T(1, x) = x => T(1,2) = 2 is prime => a(0) = 2;
T(2, x) = 2x^2 - 1 => T(2, 2) = 7 is prime => a(1) = 2;
T(4, x) = 8x^4 - 8x^2 + 1 => T(4,2) = 97 is prime => a(2) = 2.

Cf. A066436, A144131, A144132, A219276, A219277, A219278, A219279, A219280.

for n from 0 to 11 do
  P:= unapply(orthopoly[T](2^n,x),x):
  for k from 1 do if isprime(P(k)) then A[n]:= k; break fi od
od:
seq(A[n],n=0..11); # Robert Israel, Aug 13 2018

Table[k = 0; While[!PrimeQ[ChebyshevT[2^n,k]], k++]; k, {n, 0, 7}]

a(10) and a(11) from Robert Israel, Aug 13 2018

cp's OEIS Frontend

A331265 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31^2)^2 = y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A124142 Abundant numbers k such that sigma(k) is a perfect power.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A143834 Numbers k such that 2k^2 - 1 is not prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

A219280 Smallest prime of the form ChebyshevT[2^n, x].

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

A242930 Primes of the form (k^2+7)/11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

A256917 Primes which are not the sums of two consecutive nonsquares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A219281 Smallest number k such that ChebyshevT[2^n, k] is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions