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.

A028916 Friedlander-Iwaniec primes: Primes of form a^2 + b^4.

Original entry on oeis.org

2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241, 257, 277, 281, 337, 401, 457, 577, 617, 641, 661, 677, 757, 769, 821, 857, 881, 977, 1097, 1109, 1201, 1217, 1237, 1297, 1301, 1321, 1409, 1481, 1601, 1657, 1697, 1777, 2017, 2069, 2137, 2281, 2389, 2417, 2437
Offset: 1

Views

Author

Warut Roonguthai

Keywords

Comments

John Friedlander and Henryk Iwaniec proved that there are infinitely many such primes.
A256852(A049084(a(n))) > 0. - Reinhard Zumkeller, Apr 11 2015
Primes in A111925. - Robert Israel, Oct 02 2015
Its intersection with A185086 is A262340, by the uniqueness part of Fermat's two-squares theorem. - Jonathan Sondow, Oct 05 2015
Cunningham calls these semi-quartan primes. - Charles R Greathouse IV, Aug 21 2017
Primes of the form (x^2 + y^2)/2, where x > y > 0, such that (x-y)/2 or (x+y)/2 is square. - Thomas Ordowski, Dec 04 2017
Named after the Canadian mathematician John Benjamin Friedlander (b. 1941) and the Polish-American mathematician Henryk Iwaniec (b. 1947). - Amiram Eldar, Jun 19 2021

Examples

			2 = 1^2 + 1^4.
5 = 2^2 + 1^4.
17 = 4^2 + 1^4 = 1^2 + 2^4.

Links

Crossrefs

Cf. A078523, A111925.
Cf. A000290, A000583, A000040, A256852, A256863 (complement), A002645 (subsequence), subsequence of A247857.
Primes of form n^2 + b^4, b fixed: A002496 (b = 1), A243451 (b = 2), A256775 (b = 3), A256776 (b = 4), A256777 (b = 5), A256834 (b = 6), A256835 (b = 7), A256836 (b = 8), A256837 (b = 9), A256838 (b = 10), A256839 (b = 11), A256840 (b = 12), A256841 (b = 13).

Programs

  • Haskell
    a028916 n = a028916_list !! (n-1)
a028916_list = map a000040 $ filter ((> 0) . a256852) [1..]
-- Reinhard Zumkeller, Apr 11 2015
  • Maple
    N:= 10^5: # to get all terms <= N
S:= {seq(seq(a^2+b^4, a = 1 .. floor((N-b^4)^(1/2))),b=1..floor(N^(1/4)))}:
sort(convert(select(isprime,S),list)); # Robert Israel, Oct 02 2015
  • Mathematica
    nn = 10000; t = {}; Do[n = a^2 + b^4; If[n <= nn && PrimeQ[n], AppendTo[t, n]], {a, Sqrt[nn]}, {b, nn^(1/4)}]; Union[t] (* T. D. Noe, Aug 06 2012 *)
  • PARI
    list(lim)=my(v=List([2]),t);for(a=1,sqrt(lim\=1),forstep(b=a%2+1, sqrtint(sqrtint(lim-a^2)), 2, t=a^2+b^4;if(isprime(t),listput(v,t)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jun 12 2013

Extensions

Title expanded by Jonathan Sondow, Oct 02 2015

A185086 Fouvry-Iwaniec primes: Primes of the form k^2 + p^2 where p is a prime.

Original entry on oeis.org

5, 13, 29, 41, 53, 61, 73, 89, 109, 113, 137, 149, 157, 173, 193, 229, 233, 269, 281, 293, 313, 317, 349, 353, 373, 389, 397, 409, 433, 449, 461, 509, 521, 557, 569, 593, 601, 613, 617, 653, 673, 701, 733, 761, 773, 797, 809, 853, 857, 877, 929, 937, 941, 953
Offset: 1

Views

Author

Charles R Greathouse IV, Feb 18 2011

Keywords

Comments

Sequence is infinite, see Fouvry & Iwaniec.
Its intersection with A028916 is A262340, by the uniqueness part of Fermat's two-squares theorem. - Jonathan Sondow, Oct 05 2015
Named after the French mathematician Étienne Fouvry (b. 1953) and the Polish-American mathematician Henryk Iwaniec (b. 1947). - Amiram Eldar, Jun 20 2021

Links

Crossrefs

Subsequence of A002144 and hence of A002313.
The positive terms of A240130 form a subsequence.
Cf. A010052, A001248, A000040, A028916, A262340.

Programs

  • Haskell
    a185086 n = a185086_list !! (n-1)
a185086_list = filter (\p -> any ((== 1) . a010052) $
               map (p -) $ takeWhile (<= p) a001248_list) a000040_list
-- Reinhard Zumkeller, Mar 17 2013
  • Mathematica
    nn = 1000; Union[Reap[Do[n = k^2 + p^2; If[n <= nn && PrimeQ[n], Sow[n]], {k, Sqrt[nn]}, {p, Prime[Range[PrimePi[Sqrt[nn]]]]}]][[2, 1]]]
  • PARI
    is(n)=forprime(p=2,sqrtint(n),if(issquare(n-p^2),return(isprime(n))));0
  • PARI
    list(lim)=my(v=List(),N,t);forprime(p=2,sqrt(lim), N=p^2; for(n=1,sqrt(lim-N), if(ispseudoprime(t=N+n^2), listput(v,t)))); v=vecsort(Vec(v),,8); v

A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

Original entry on oeis.org

2, 3, 5, 5, 7, 5, 3, 5, 3, 7, 11, 7, 11, 13, 7, 2, 13, 13, 5, 17, 13, 11, 5, 17, 7, 17, 19, 3, 17, 7, 19, 5, 11, 19, 13, 23, 5, 17, 19, 13, 23, 5, 2, 19, 17, 11, 5, 23, 29, 29, 23, 19, 29, 13, 31, 31, 23, 11, 3, 5, 31, 13, 2, 29, 5, 13, 31, 2, 11, 5, 31, 37, 23, 37, 3, 7, 23, 3, 13, 31, 19, 37, 41, 11
Offset: 1

Views

Author

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

Keywords

Comments

The sequence is well-defined by the uniqueness part of Fermat's two-squares theorem.
The sequence is infinite, since Fouvry and Iwaniec proved that A185086 is infinite.

Examples

			A185086(2) = 13 = 2^2 + 3^2 and A185086(6) = 61 = 5^2 + 6^2, so a(2) = 3 and a(6) = 5.

Links

Crossrefs

Cf. A002313, A002144, A028916, A045637, A062324, A185086, A240130, A262340, A263722.

Programs

  • Mathematica
    p = 2; lst = {}; While[p < 100, k = 1; While[k < 101, If[PrimeQ[k^2 + p^2], AppendTo[lst, {k^2 + p^2, If[PrimeQ@ k, Max[k, p], p]}]]; k++]; p = NextPrime@ p]; Transpose[Union@ lst][[2]]
  • PARI
    do(lim)=my(v=List(),p2,t); forprime(p=2, sqrtint(lim\=1), p2=p^2; for(k=1, sqrtint(lim-p2), if(isprime(t=p2+k^2), listput(v, [t, if(isprime(k),max(k,p),p)])))); v=vecsort(Set(v),1); apply(u->u[2], v) \\ Charles R Greathouse IV, Aug 21 2017

Formula

a(n)^2 = A185086(n) - k^2 for some integer k > 0.
Showing 1-3 of 3 results.

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