A078523 -id:A078523 - cp's OEIS frontend

A331522 a(n) is the least positive k such that A078523(n) - k^2 is a sixth power.

1, 2, 4, 6, 3, 5, 10, 7, 14, 13, 16, 17, 20, 24, 23, 26, 2, 10, 20, 33, 22, 35, 36, 37, 40, 32, 43, 45, 47, 54, 55, 56, 50, 57, 52, 58, 9, 11, 65, 66, 19, 25, 29, 73, 74, 75, 45, 84, 55, 80, 59, 61, 90, 93, 94, 71, 97, 75, 98, 79, 81, 100, 85, 110, 91, 110, 95

Offset: 1

Rémy Sigrist, Jan 19 2020

nonn

			The first terms, alongside A078523(n), are:
  n   a(n)  A078523(n)
  --  ----  ----------------
   1     1    2 =  1^2 + 1^6
   2     2    5 =  2^2 + 1^6
   3     4   17 =  4^2 + 1^6
   4     6   37 =  6^2 + 1^6
   5     3   73 =  3^2 + 2^6
   6     5   89 =  5^2 + 2^6
   7    10  101 = 10^2 + 1^6
   8     7  113 =  7^2 + 2^6
   9    14  197 = 14^2 + 1^6
  10    13  233 = 13^2 + 2^6

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A331522

See A331435 for similar sequences.

Cf. A001014, A078523.

PARI
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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

Warut Roonguthai

nonn

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

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

T. D. Noe, Table of n, a(n) for n = 1..10000
Art of Problem Solving, Fermat's Two Squares Theorem.
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics, Vol. 36 (1907), pp. 145-174.
John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, Proc. Nat. Acad. Sci., Vol. 94 (1997), pp. 1054-1058.
J. Friedlander and H. Iwaniec, The polynomial x^2 + y^4 captures its primes, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.
Charles R Greathouse IV, Tables of special primes.
Wikipedia, Friedlander-Iwaniec theorem.

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).

a028916 n = a028916_list !! (n-1)
a028916_list = map a000040 $ filter ((> 0) . a256852) [1..]
-- Reinhard Zumkeller, Apr 11 2015

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

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 *)

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

Title expanded by Jonathan Sondow, Oct 02 2015

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A331522 a(n) is the least positive k such that A078523(n) - k^2 is a sixth power.

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

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