A226495 -id:A226495 - cp's OEIS frontend

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

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

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

nonn

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.

			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.

Cf. A028916, A226495, A226497, A226498.

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}]

A226497 The number of primes of the form i^2+j^4 (A028916) <= 10^n.

Original entry on oeis.org

2, 6, 28, 121, 583, 2724, 13175, 64551, 322110, 1621929, 8254127

Offset: 1

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

Even if a prime has more than one representation in the form i^2+j^4, it is only counted once.

Iwaniec and Friedlander have proved there is infinity of the primes of the form i^2+j^4.

Cf. A028916, A226495, A226496 & A226498.

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, # < 10^n &], {n, 12}]

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

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

nonn

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.

John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS February 18, 1997 94 (4) 1054-1058.

Cf. A028916, A226495, A226496, A226497.

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}]

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

cp's OEIS Frontend

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

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A226497 The number of primes of the form i^2+j^4 (A028916) <= 10^n.

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

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