A049002 -id:A049002 - cp's OEIS frontend

A078597 Primes of the form p*(p+4)+2 where p and p+4 are primes.

23, 79, 223, 439, 4759, 53359, 77839, 95479, 99223, 159199, 194479, 239119, 378223, 416023, 680623, 2223079, 2595319, 2873023, 3186223, 3515623, 4003999, 5022079, 6456679, 6859159, 8732023, 9235519, 9492559, 10017223, 10595023

Offset: 1

Cino Hilliard, Dec 08 2002

More generally, if a and b are even numbers, let Seq(a,b) be the sequence of primes of the form p*(p+a)+b where p and p+a are primes. Seq(a,b) is finite if either a^2+b == 2 (mod 3) or a^2-4*b is a square. Is it infinite in all other cases?

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Except for the term 23, this is a subsequence of A048880. A051779 is Seq(2, 2). A049002 is Seq(0, -2). A045637 is Seq(0, 4).

Select[ #(#+4)+2&/@Select[Prime/@Range[500], PrimeQ[ #+4]&], PrimeQ]

prodtp(n1,n2,a,b)=local(f,x); f=0; forprime(x=n1,n2,if(isprime(x+a),f=x*(x+a)+b; if(isprime(f),print(x" "x+a" "f" "); ); ); ); \ Computes that part of Seq(a,b) with n1<=p<=n2.

Edited by Dean Hickerson, Dec 10 2002

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

Original entry on oeis.org

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

Giovanni Teofilatto, Nov 04 2008

nonn

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

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

Cf. A013929, A049002.

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

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 *)

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

[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

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

A182474 Primes of the form p^q - q, where p and q are primes.

Original entry on oeis.org

2, 5, 7, 23, 47, 167, 359, 839, 1367, 1847, 2207, 3719, 5039, 7919, 8179, 10607, 11447, 16127, 17159, 19319, 29927, 36479, 44519, 49727, 54287, 57119, 66047, 85847, 97967, 113567, 128879, 177239, 196247, 201599, 218087, 241079, 273527, 292679, 323759, 344567

Offset: 1

Alex Ratushnyak, May 01 2012

			8179 = 2^13 - 13

Cf. A049002 (primes of the form p^2 - 2).

Cf. A057678 (primes of the form 2^p - p).

nn = 600000; mx = Floor[Log[2, nn]]; t2 = Select[Table[2^n - n, {n, Prime[Range[PrimePi[mx]]]}], PrimeQ]; mx = Floor[Sqrt[nn]]; tp = Select[Table[n^2 - 2, {n, Prime[Range[PrimePi[mx]]]}], PrimeQ]; Union[t2, tp] (* T. D. Noe, May 01 2012 *)
Module[{upto=350000,r},r=Floor[Sqrt[upto+2]];Select[Union[Select[ (#1[[1]]^#1[[2]]-#1[[2]]&)/@Tuples[Prime[Range[r]],2], PrimeQ]], #1<=upto&]] (* Harvey P. Dale, Dec 07 2012 *)

Union of A049002 and A057678.

cp's OEIS Frontend

A078597 Primes of the form p*(p+4)+2 where p and p+4 are primes.

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

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A182474 Primes of the form p^q - q, where p and q are primes.

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