A078667 Integers that occur more than once as the difference of the squares of two consecutive primes.

72, 120, 168, 312, 408, 552, 600, 768, 792, 912, 1032, 1848, 2472, 3048, 3192, 3288, 3528, 3720, 4008, 4920, 5160, 5208, 5808, 5928, 6072, 6480, 6792, 6840, 6888, 7080, 7152, 7248, 7512, 7728, 7800, 8520, 8760, 9072, 11400, 11880, 11928, 12792, 13200, 13320

Offset: 1

Author

Jon Perry, Dec 15 2002

Keywords

• nonn

Comments

1848 is the first integer that occurs exactly three times. The next few are 6888, 14280, 16008, 19152. 4920 is the first integer that occurs exactly four times. See A069482 for more details. - Richard R. Forberg, Feb 06 2015

Examples

			120 = 31^2 - 29^2 = 17^2 - 13^2.
		

Links

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

Crossrefs

Cf. A090783, A090784, A090785, A091878.

Programs

Maple

N:= 20000: # for terms <= N
V:= Vector(N):
p:= 3:
while 4*(p-1) <= N do
  q:= p; p:= nextprime(p);
  v:= p^2 - q^2;
  if v > N then next fi;
  V[v]:= V[v]+1
od:
select(v -> V[v] > 1, 2*[$1..N/2]); # Robert Israel, Aug 22 2025

Mathematica

Sort@ DeleteDuplicates@ Flatten@ Select[Gather[NextPrime[#]^2 - #^2 & /@ Prime@ Range@ 1200], Length@ # > 1 &] (* Michael De Vlieger, Mar 18 2015 *)
Select[Tally[Differences[Prime[Range[1000]]^2]],#[[2]]>1&][[;;,1]]//Sort (* Harvey P. Dale, Nov 16 2023 *)

PARI

pv(v)=vecsort(vecextract(v,concat("1..",vc-1))) op=2; v=vector(5000); vc=1; forprime (p=3,5000,v[vc]=p^2-op^2; vc++; op=p) v=pv(v) for (i=2,length(v), if (v[i]==v[i-1],print1(v[i]",")))

Extensions

Duplicate terms removed, as suggested by Richard R. Forberg, by Jon E. Schoenfield, Mar 15 2015