A078667 -id:A078667 - cp's OEIS frontend

A090781 Numbers that can be expressed as the difference of the squares of primes in just one distinct way.

5, 16, 21, 24, 40, 45, 48, 96, 112, 117, 144, 160, 165, 192, 264, 280, 285, 288, 336, 352, 357, 504, 520, 525, 648, 816, 832, 837, 936, 952, 957, 1152, 1344, 1360, 1365, 1368, 1440, 1656, 1672, 1677, 1752, 1824, 1840, 1845, 1872, 1968, 2184, 2200, 2205, 2328

Offset: 1

Ray Chandler, Feb 14 2004

nonn

			5 = 3^2 - 2^2.

T. D. Noe, Table of n, a(n) for n=1..908

Cf. A078667, A090788, A090782, A090785.

With[{nn=100},Take[Sort[Transpose[Select[Tally[Last[#]-First[#]&/@ Subsets[ Prime[Range[nn]]^2,{2}]],Last[#]==1&]][[1]]],nn]] (* Harvey P. Dale, Apr 05 2014 *)

from sympy import primerange
from collections import Counter
def aupto(limit):
  sqps = [p*p for p in primerange(1, limit//2+1)]
  ways = Counter(b-a for i, a in enumerate(sqps) for b in sqps[i+1:])
  return sorted(k for k in ways if k <= limit and ways[k] == 1)
print(aupto(2328)) # Michael S. Branicky, May 16 2021

A090782 Numbers that can be expressed as the difference of the squares of primes in just three distinct ways.

Original entry on oeis.org

120, 168, 312, 408, 480, 552, 600, 672, 720, 1008, 1200, 1800, 2160, 2472, 2832, 2880, 3312, 3672, 4560, 5040, 5640, 6120, 6480, 6528, 7248, 7320, 7752, 7872, 8160, 8352, 8400, 8712, 8880, 9048, 9768, 9960, 10032, 10200, 10320, 10488, 10608, 10848

Offset: 1

Ray Chandler, Feb 14 2004

nonn

			120 = 13^2-7^2 = 17^2-13^2 = 31^2-29^2.

Cf. A078667, A090788, A090781.

A090788 Numbers that can be expressed as the difference of the squares of primes in just two distinct ways.

Original entry on oeis.org

72, 360, 432, 528, 768, 888, 960, 1032, 1080, 1128, 1272, 1392, 1488, 1512, 1608, 1632, 1728, 1920, 2088, 2112, 2232, 2352, 2400, 2448, 2568, 2688, 2808, 3048, 3168, 3240, 3288, 3480, 3648, 3768, 4008, 4032, 4128, 4248, 4272, 4392, 4488, 4512, 4992

Offset: 1

Ray Chandler, Feb 14 2004

nonn

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

Cf. A078667, A090781, A090782.

i=4; while(i<=5000, k=0; m=2; while(m*m<=i, if(i%(2*m)==0, a=(i/m-m)/2; b=a+m; if(isprime(a)&&isprime(b), k+=1)); m+=2); if(k==2, print1(i, ", ")); i+=4) \\ Antonio Roldán, Nov 06 2018

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

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

Ray G. Opao, Feb 08 2004

nonn

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

Cf. A078667, A090783, A090785, A091878.

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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A090781 Numbers that can be expressed as the difference of the squares of primes in just one distinct way.

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A090782 Numbers that can be expressed as the difference of the squares of primes in just three distinct ways.

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A090788 Numbers that can be expressed as the difference of the squares of primes in just two distinct ways.

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

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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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