A090784 -id:A090784 - cp's OEIS frontend

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

Jon Perry, Dec 15 2002

nonn

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

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

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

Cf. A090783, A090784, A090785, A091878.

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

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 *)

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]",")))

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

A090783 a(n) can be expressed as the difference of the squares of consecutive primes in just three distinct ways.

Original entry on oeis.org

1848, 6888, 14280, 16008, 19152, 36120, 112728, 116832, 129480, 139080, 176520, 190632, 190968, 199752, 216840, 236208, 252120, 274848, 303960, 314160, 340368, 363720, 435792, 458280, 503160, 513240, 686160, 688680, 698880, 712680, 721560

Offset: 1

Ray G. Opao, Feb 08 2004

nonn

			1848 = 463^2 - 461^2 = 233^2 - 229^2 = 157^2 - 151^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 3091 terms from Robert Israel)

Cf. A078667, A090784, A090785, A091878.

N:= 10^6: # to get all terms <= N
V:= Vector(N/4):
p:= 3:
while p < N/2 do
  q:= p;
  p:= nextprime(p);
  r:= (p^2-q^2)/4;
  if r <= N/4 then
    V[r]:= V[r]+1
  fi
od:
map(`*`,select(t -> V[t]=3, [$1..N/4]),4); # Robert Israel, Aug 13 2018

is(n) = my(i=0, v=[]); forprime(p=5, n, v=[precprime(p-1), p]; if(v[2]^2-v[1]^2==n, i++)); i==3 \\ Felix Fröhlich, Aug 13 2018

More terms from Ray Chandler, Feb 11 2004

A090785 Numbers that can be expressed as the difference of the squares of consecutive primes in just one way.

Original entry on oeis.org

5, 16, 24, 48, 240, 288, 360, 432, 648, 672, 720, 840, 888, 960, 1080, 1128, 1248, 1320, 1392, 1488, 1560, 1608, 1680, 1728, 1800, 1920, 2040, 2088, 2112, 2232, 2280, 2400, 2520, 2568, 2640, 2808, 2832, 2880, 3120, 3240, 3312, 3360, 3432, 3672, 3912, 4080

Offset: 1

Ray G. Opao, Feb 08 2004

nonn

			5 = 3^2 - 2^2.

Cf. A078667, A090781, A090783, A090784, A091878.

from sympy import primerange
from collections import Counter
def aupto(limit):
  sqps = [p*p for p in primerange(1, limit//2+1)]
  ways = Counter(sqps[i+1]-sqps[i] for i in range(len(sqps)-1))
  return sorted(k for k in ways if k <= limit and ways[k] == 1)
print(aupto(4080)) # Michael S. Branicky, May 16 2021

More terms from Ray Chandler, Feb 11 2004

A091878 a(n) can be expressed as the difference of the squares of consecutive primes in just four distinct ways.

Original entry on oeis.org

4920, 801528, 1484280, 5351640, 7257720, 7647360, 12168912, 12302472, 15540360, 17348520, 21623160, 25639320, 27285048, 27462840, 27470352, 34714680, 35684040, 38466288, 48449640, 49936152, 51272760, 54037368, 60948888

Offset: 1

Ray G. Opao and Ray Chandler, Feb 11 2004

nonn

			4920=211^2-199^2=251^2-241^2=617^2-613^2=1231^2-1229^2

Cf. A078667, A090783, A090784, A090785.

#define NMAX 200000000 #include  #include  #include  int isprime(const int n) { for(int i=2 ; i*i <= n ; i++) if( n % i == 0) return(0) ; return 1 ; } int main(int argc, char *argv[]) { short * n= (short*)calloc(NMAX,sizeof(short)) ; int wm=0; for(int p=2 ; ; ) { int np=p+1 ; while( !isprime(np) ) np++ ; if(np<0) break ; if ( p+np < INT_MAX/(np-p) ) { const int i=(p+np)*(np-p) ; const int nw= p+np ; if( i < NMAX ) n[i]++ ; for(int j=wm ; j < nw ; j++) if ( n[j] == 4) printf("%d ",j) ; wm=nw ; } p=np ; if( p > INT_MAX-np ) break ; } free(n) ; } - R. J. Mathar, Oct 05 2006

More terms from R. J. Mathar, Oct 05 2006

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

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A090783 a(n) can be expressed as the difference of the squares of consecutive primes in just three distinct ways.

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A090785 Numbers that can be expressed as the difference of the squares of consecutive primes in just one way.

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A091878 a(n) can be expressed as the difference of the squares of consecutive primes in just four distinct ways.

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