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 1-3 of 3 results.

A226495 The number of primes of the form i^2+j^4 (A028916) <= 10^n, counted with multiplicity.

Original entry on oeis.org

2, 8, 34, 134, 615, 2813, 13415, 65162, 323858, 1626844, 8268241, 42417710
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Iwaniec and Friedlander have proved there is infinity of the primes of the form i^2+j^4.
Primes with more than one representation are counted multiple times.
If we do not count repetitions, the sequence is A226497: 2, 6, 28, 121, 583, 2724, 13175, 64551, ..., .

Examples

			2 = 1^2+1^4, 5 = 2^2+1^4, 17 = 4^2+1^4 = 1^2+2^4, …, 97 = 9^2+2^4 = 4^2+3^4, etc.
		

Crossrefs

Programs

  • Mathematica
    mx = 10^12; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[ lst, a]], {i, Sqrt[mx]}, {j, Sqrt[ Sqrt[mx - i^2]]}]; Table[ Length@ Select[lst, # < 10^n &], {n, 12}]

A226496 The number of primes of the form i^2 + j^4 (A028916) <= 2^n, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 9, 13, 21, 34, 50, 77, 121, 191, 292, 458, 727, 1164, 1840, 2904, 4650, 7429, 11869, 19087, 30760, 49474, 79971, 129226, 209823, 340347, 552722, 898655, 1461698, 2381041, 3883079, 6338935, 10357549, 16935173, 27712338, 45381521, 51559329
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Iwaniec and Friedlander have proved there is infinity of the primes of the form i^2 + j^4.
Counted with double representations. If we do not count doubles, the sequence is A226498.

Examples

			2 = 1^2+1^4, 5 = 2^2+1^4, 17 = 4^2+1^4 = 1^2+2^4, ..., 97 = 9^2+2^4 = 4^2+3^4, etc.
		

Crossrefs

Programs

  • Mathematica
    mx = 2^40; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[ lst, a]], {i, Sqrt[mx]}, {j, Sqrt[ Sqrt[mx - i^2]]}]; Table[ Length@ Select[lst, # <2^n &], {n, 40}]

A226498 The number of primes of the form i^2 + j^4 (A028916) <= 2^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 7, 11, 17, 28, 43, 67, 108, 173, 272, 434, 690, 1115, 1772, 2815, 4528, 7267, 11646, 18799, 30378, 48956, 79270, 128267, 208509, 338533, 550262, 895284, 1457111, 2374753, 3874445, 6327042
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Iwaniec and Friedlander proved there are infinity of the primes of the form i^2+j^4, and hence a(n) increases without bound.
Does not count double representations.

Crossrefs

Programs

  • Mathematica
    mx = 2^40; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[lst, a]], {i, Sqrt[mx]}, {j, Sqrt[ Sqrt[mx - i^2]]}]; Table[ Length@ Select[ Union@ lst, # < 2^n &], {n, 40}]
  • PARI
    a(n)=my(N=2^n,v=List(),t);for(a=1,sqrt(N),forstep(b=a%2+1, sqrtint(sqrtint(N-a^2)), 2, t=a^2+b^4;if(isprime(t),listput(v,t)))); 1+#vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jun 12 2013
Showing 1-3 of 3 results.