A146980 Nonsquarefree numbers such that n-1 is prime and n+1 is square.

8, 24, 48, 80, 168, 224, 360, 440, 728, 840, 1088, 1224, 1368, 1848, 2208, 2400, 3024, 3720, 3968, 4760, 5040, 5624, 5928, 7920, 8648, 10608, 11448, 13688, 14160, 14640, 16128, 17160, 18224, 19320, 21024, 24024, 25920, 28560, 29928, 31328, 33488

Offset: 1

Author

Giovanni Teofilatto, Nov 04 2008

Keywords

• nonn

Comments

Also numbers n > 3 such that n-1 is prime and n+1 is square.

Sequence gives values x of fundamental solution (x,y) to Pellian x^2 - D*y^2 = 1, with D = n-1 = A049002, corresponding values y being sqrt(n+1) = A028870. (Substituting back into the Pellian we indeed have n^2 - (n-1)(n+1) = 1.) - Lekraj Beedassy, Feb 23 2019

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A013929, A049002.

Programs

Magma

[ n: n in [1..35000] | not IsSquarefree(n) and IsPrime(n-1) and IsSquare(n+1) ]; // Klaus Brockhaus, Nov 05 2008

Mathematica

Select[Range[35000], !SquareFreeQ[#] && PrimeQ[#-1] && IntegerQ[Sqrt[#+1] ] &] (* G. C. Greubel, Feb 22 2019 *)
Mean/@SequencePosition[Table[Which[PrimeQ[n],1,IntegerQ[Sqrt[ n]],3,!SquareFreeQ[ n],2,True,0],{n,33500}],{1,2,3}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 03 2020 *)

PARI

list(lim)=my(v=List()); forstep(k=3,sqrtint(lim\1+1),2, if(isprime(k^2-2), listput(v,k^2-1))); Vec(v) \\ Charles R Greathouse IV, Jun 13 2017

Sage

[n for n in (1..35000) if not is_squarefree(n) and is_prime(n-1) and is_square(n+1)] # G. C. Greubel, Feb 22 2019

Extensions

Extended beyond a(6) by Klaus Brockhaus, Nov 05 2008