cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Jon Perry, Dec 15 2002

Keywords

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

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

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

Views

Author

Ray G. Opao, Feb 08 2004

Keywords

Examples

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

Links

Crossrefs

Cf. A078667, A090784, A090785, A091878.

Programs

  • Maple
    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
  • PARI
    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

Extensions

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

Views

Author

Ray G. Opao, Feb 08 2004

Keywords

Examples

			5 = 3^2 - 2^2.

Crossrefs

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

Programs

  • Python
    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

Extensions

More terms from Ray Chandler, Feb 11 2004

A090784 Numbers that can be expressed as the difference of the squares of consecutive primes in just two distinct ways.

Original entry on oeis.org

72, 120, 168, 312, 408, 552, 600, 768, 792, 912, 1032, 2472, 3048, 3192, 3288, 3528, 3720, 4008, 5160, 5208, 5808, 5928, 6072, 6480, 6792, 6840, 7080, 7152, 7248, 7512, 7728, 7800, 8520, 8760, 9072, 11400, 11880, 11928, 12792, 13200, 13320, 13440
Offset: 1

Views

Author

Ray G. Opao, Feb 08 2004

Keywords

Examples

			72 = 19^2-17^2 = 11^2-7^2.

Crossrefs

Cf. A078667, A090783, A090785, A091878.

Extensions

More terms from Ray Chandler, Feb 11 2004
Showing 1-4 of 4 results.

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