A124865 -id:A124865 - cp's OEIS frontend

A283562 Primes of the form (p^2 - q^2) / 24 with primes p > q > 3.

2, 3, 5, 7, 11, 13, 17, 23, 37, 43, 47, 53, 67, 73, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 223, 233, 257, 263, 277, 283, 293, 313, 337, 347, 353, 373, 397, 433, 443, 467, 487, 523, 547, 563, 577, 593, 607, 613, 617, 643, 647, 653, 733, 743, 757, 773, 787, 797, 887, 907, 937, 947, 953, 977

Offset: 1

Altug Alkan and Thomas Ordowski, Mar 11 2017

nonn

Note that p - q must be <= 12. Also note that there can be corresponding prime pairs (q, p) more than one way, i.e., (7, 13), (13, 17), (29, 31): (13^2 - 7^2)/24 = (17^2 - 13^2)/24 = (31^2 - 29^2)/24 = 5.
There are no terms of A045468 > 11.
Union of {2}, A006489, A060212, A092110, and A125272. - Robert Israel, Mar 13 2017

			3 is a term since (11^2 - 7^2)/24 = 3 and 3, 7, 11 are prime numbers.

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

Cf. A006489, A060212, A092110, A124865, A124866, A125272, A283532.

select(r -> isprime(r) and ((isprime(3*r+2) and isprime(3*r-2))
  or (isprime(6*r+1) and isprime(6*r-1))
  or (isprime(2*r+3) and isprime(2*r-3))
or (isprime(r+6) and isprime(r-6))), [2,seq(i,i=3..1000,2)]); # Robert Israel, Mar 13 2017

ok[n_] := PrimeQ[n] && Block[{p, q, s = Reduce[p^2-q^2 == 24 n && p>3 && q>3, {p, q}, Integers]}, If[s === {}, False, Or @@ And @@@ PrimeQ[{p, q} /. List@ ToRules@s]]]; Select[Range@1000, ok] (* Giovanni Resta, Mar 11 2017 *)

isA124865(n) = if(n%24, isprimepower(n+4)==2 || isprimepower(n+9)==2, fordiv(n/4, d, if(isprime(n/d/4+d) && isprime(n/d/4-d), return(1))); 0)
lista(nn) = forprime(p=2, nn, if(isA124865(24*p), print1(p", ")))

For n > 5, a(n) == {3,7} mod 10.

A124866 Numbers of the form (p^2-q^2)/8, p, q odd primes, p>q.

Original entry on oeis.org

2, 3, 5, 6, 9, 12, 14, 15, 18, 20, 21, 24, 30, 33, 35, 36, 39, 42, 44, 45, 51, 54, 60, 63, 65, 66, 69, 75, 81, 84, 90, 96, 99, 102, 104, 105, 111, 114, 117, 119, 120, 126, 129, 135, 141, 144, 150, 156, 159, 165, 168, 170, 171, 174, 180, 186, 189, 195, 201, 204, 207

Offset: 1

Alexander Adamchuk, Nov 10 2006

nonn

Primes in a(n) are {2, 3, 5}.

Cf. A124865 Numbers of the form p^2-q^2, p, q primes, p>q. Cf. A045636 Numbers of the form p^2+q^2, p, q primes.

Take[Union[Flatten[Table[(Prime[p]^2 - Prime[q]^2)/8, {p, 2, 100}, {q, 2, p - 1}]]], 60] (* Alonso del Arte, Jul 14 2011 *)

A358313 Primes p such that 24*p is the difference of two squares of primes in three different ways.

Original entry on oeis.org

5, 7, 13, 17, 23, 103, 6863, 7523, 11807, 11833, 22447, 91807, 100517, 144167, 204013, 221077, 478937, 531983, 571867, 752293, 1440253, 1647383, 1715717, 1727527, 1768667, 2193707, 2381963, 2539393, 2957237, 3215783, 3290647, 3873713, 4243997, 4512223, 4880963, 4895777, 5226107, 5345317, 5540063

Offset: 1

J. M. Bergot and Robert Israel, Nov 08 2022

nonn

The positive integer solutions of 24*p = x^2 - y^2 are (x = p+6, y = p-6), (x = 2*p+3, y = 2*p - 3), (x = 3*p+2, y = 3*p-2) and (x = 6*p+1, y=6*p-1). Since at least one of these is always divisible by 7, it is impossible for 24*p to be the difference of two squares of primes in 4 different ways.
Primes p such that three of the pairs (p +- 6), (2*p +- 3), (3*p +- 2), (6*p +- 1) are pairs of primes.
Except for 5, all terms == 3 or 7 (mod 10).

			a(3) = 13 is a term because 13 is prime, 13 +- 6 = 19 and 7 are primes, 2*13 +- 3 = 29 and 23 are primes, and 3*13 +- 2 = 37 and 41 are primes.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A124865.

filter:= proc(p) local t;
  if not isprime(p) then return false fi;
  t:= 0;
  if isprime(p+6) and isprime(p-6) then t:= t+1 fi;
  if isprime(2*p+3) and isprime(2*p-3) then t:= t+1 fi;
  if isprime(3*p+2) and isprime(3*p-2) then t:= t+1 fi;
  if isprime(6*p+1) and isprime(6*p-1) then t:= t+1 fi;
  t = 3
end proc:
select(filter, [seq(i,i=3..10^7,2)]);

A132329 Numbers that cannot be expressed as the difference of the squares of primes.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Rick L. Shepherd, Aug 19 2007

nonn

Complement of A124865.

Cf. A124865, A090781.

lim=80;sp={4};i=2;Until[Prime[i]^2-(Prime[i]-2)^2>lim,AppendTo[sp,Prime[i]^2];i++];Complement[Range[lim],Union[Differences/@Subsets[sp,{2}]//Flatten]] (* James C. McMahon, Mar 08 2025 *)

A168469 Numbers of the form p^2 - q^2, p, q primes, p > q, with repetitions.

Original entry on oeis.org

5, 16, 21, 24, 40, 45, 48, 72, 72, 96, 112, 117, 120, 120, 120, 144, 160, 165, 168, 168, 168, 192, 240, 240, 240, 240, 264, 280, 285, 288, 312, 312, 312, 336, 352, 357, 360, 360, 408, 408, 408, 432, 432, 480, 480, 480, 504, 520, 525, 528, 528, 552, 552, 552

Offset: 1

Zak Seidov, Nov 26 2009

nonn

Cf. A124865 Numbers of the form p^2 - q^2, p, q primes, p > q.

Take[Sort[Last[#]^2-First[#]^2&/@Subsets[Prime[Range[40]],{2}]],60] (* Harvey P. Dale, Jul 22 2012 *)

cp's OEIS Frontend

A283562 Primes of the form (p^2 - q^2) / 24 with primes p > q > 3.

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A124866 Numbers of the form (p^2-q^2)/8, p, q odd primes, p>q.

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A358313 Primes p such that 24*p is the difference of two squares of primes in three different ways.

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A132329 Numbers that cannot be expressed as the difference of the squares of primes.

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A168469 Numbers of the form p^2 - q^2, p, q primes, p > q, with repetitions.

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