A028916 -id:A028916 - cp's OEIS frontend

A331435 a(n) is the least positive k such that A028916(n) - k^2 is a fourth power.

1, 2, 1, 6, 5, 4, 10, 11, 10, 14, 15, 1, 14, 5, 9, 20, 21, 24, 19, 4, 6, 26, 26, 12, 14, 29, 16, 31, 29, 22, 24, 31, 34, 1, 26, 5, 28, 35, 40, 19, 41, 39, 44, 38, 29, 45, 42, 4, 6, 35, 51, 16, 46, 20, 51, 54, 55, 56, 56, 30, 52, 54, 34, 36, 56, 58, 40, 9, 11

Offset: 1

Rémy Sigrist, Jan 18 2020

nonn

			The first terms, alongside A028916(n), are:
  n   a(n)  A028916(n)
  --  ----  ----------------
   1     1    2 =  1^2 + 1^4
   2     2    5 =  2^2 + 1^4
   3     1   17 =  1^2 + 2^4
   4     6   37 =  6^2 + 1^4
   5     5   41 =  5^2 + 2^4
   6     4   97 =  4^2 + 3^4
   7    10  101 = 10^2 + 1^4
   8    11  137 = 11^2 + 2^4
   9    10  181 = 10^2 + 3^4
  10    14  197 = 14^2 + 1^4

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A331435
Wikipedia, Friedlander-Iwaniec theorem

See A002331, A331521, A331522, A331523, A331524, A331525, A331526 and A331527 for similar sequences.

Cf. A000583, A028916.

PARI
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See Links section.
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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

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

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, ..., .

			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, A226496, A226497, A226498.

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

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

A002496 Primes of the form k^2 + 1.

Original entry on oeis.org

2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177

Offset: 1

N. J. A. Sloane

It is conjectured that this sequence is infinite, but this has never been proved.
An equivalent description: primes of form P = (p1*p2*...*pm)^k + 1 where p1..pm are primes and k > 1, since then k must be even for P to be prime.
Also prime = p(n) if A054269(n) = 1, i.e., quotient-cycle-length = 1 in continued fraction expansion of sqrt(p). - Labos Elemer, Feb 21 2001
Also primes p such that phi(p) is a square.
Also primes of form x*y + z, where x, y and z are three successive numbers. - Giovanni Teofilatto, Jun 05 2004
It is a result that goes back to Mirsky that the set of primes p for which p-1 is squarefree has density A, where A = A005596 denotes the Artin constant. More precisely, Sum_{p <= x} mu(p-1)^2 = A*x/log x + o(x/log x) as x tends to infinity. Conjecture: Sum_{p <= x, mu(p-1)=1} 1 = (A/2)*x/log x + o(x/log x) and Sum_{p <= x, mu(p-1)=-1} 1 = (A/2)*x/log x + o(x/log x). - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003
Also primes of the form x^y + 1, where x > 0, y > 1. Primes of the form x^y - 1 (x > 0, y > 1) are the Mersenne primes listed in A000668(n) = {3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...}. - Alexander Adamchuk, Mar 04 2007
With the exception of the first two terms {2,5}, the continued fraction (1 + sqrt(p))/2 has period 3. - Artur Jasinski, Feb 03 2010
With the exception of the first term {2}, congruent to 1 (mod 4). - Artur Jasinski, Mar 22 2011
With the exception of the first two terms, congruent to 1 or 17 (mod 20). - Robert Israel, Oct 14 2014
From Bernard Schott, Mar 22 2019: (Start)
These primes are the primitive terms which generate the sequence of integers with only one prime factor and whose Euler's totient is a square: A054755. So this sequence is a subsequence of A054755 and of A039770. Additionally, the terms of this sequence also have a square cototient, so this sequence is a subsequence of A063752 and A054754.
If p prime = n^2 + 1, phi(p) = n^2 and cototient(p) = 1^2.
Except for 3, the four Fermat primes in A019434 {5, 17, 257, 65537}, belong to this sequence; with F_k = 2^(2^k) + 1, phi(F_k) = (2^(2^(k-1)))^2.
See the file "Subfamilies and subsequences" (& I) in A039770 for more details, proofs with data, comments, formulas and examples. (End)
In this sequence, primes ending with 7 seem to appear twice as often as primes ending with 1. This is because those with 7 come from integers ending with 4 or 6, while those with 1 come only from integers ending with 0 (see De Koninck & Mercier reference). - Bernard Schott, Nov 29 2020
The set of odd primes p for which every elliptic curve of the form y^2 = x^3 + d*x has order p-1 over GF(p) for those d with (d,p)=1 and d a fourth power modulo p. - Gary Walsh, Sep 01 2021 [edited, Gary Walsh, Apr 26 2025]

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 211 pp. 34 and 169, Ellipses, Paris, 2004.
Leonhard Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 22.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
Hugh L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
C. Stanley Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 134.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tewodros Amdeberhan, Luis A. Medina and Victor H. Moll, Arithmetical properties of a sequence arising from an arctangent sum, J. Numb. Theory, Vol. 128, No. 6 (2008), pp. 1807-1846, eq. (1.10).
William D. Banks, John B. Friedlander, Carl Pomerance and Igor E. Shparlinski, Multiplicative structure of values of the Euler function, in High Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams (A. Van der Poorten, ed.), Fields Inst. Comm. 41 (2004), pp. 29-47.
Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, Vol. 16, No. 79 (1962), pp. 363-367.
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See p. 11.
Frank Ellermann, Primes of the form (m^2)+1 up to 10^6.
Dan Ismailescu and Yunkyu James Lee, Polynomially growing integer sequences all whose terms are composite, arXiv:2501.04851 [math.NT], 2025. See p. 9.
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, Amer. Math. Monthly, Vol. 56, No. 1 (1949), pp. 17-19.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Apoloniusz Tyszka and Sławomir Kurpaska, Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X, (2020).
Eric Weisstein's World of Mathematics, Landau's Problems.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Wikipedia, Bateman-Horn Conjecture.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A083844 (number of these primes < 10^n), A199401 (growth constant).
Cf. A001912, A005574, A054964, A062325, A088179, A090693, A141293, A172168.
Cf. A000668 (Mersenne primes), A019434 (Fermat primes).
Subsequence of A039770.
Cf. A010051, subsequence of A002522.
Cf. A237040 (an analog for n^3 + 1).
Cf. A010051, A000290; subsequence of A028916.
Subsequence of A039770, A054754, A054755, A063752.
Primes of form n^2+b^4, b fixed: 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).
Cf. A030430 (primes ending with 1), A030432 (primes ending with 7).

a002496 n = a002496_list !! (n-1)
a002496_list = filter ((== 1) . a010051') a002522_list
-- Reinhard Zumkeller, May 06 2013

[p: p in PrimesUpTo(100000)| IsSquare(p-1)]; // Vincenzo Librandi, Apr 09 2011

select(isprime, [2, seq(4*i^2+1, i= 1..1000)]); # Robert Israel, Oct 14 2014

Select[Range[100]^2+1, PrimeQ]
Join[{2},Select[Range[2,300,2]^2+1,PrimeQ]] (* Harvey P. Dale, Dec 18 2018 *)

isA002496(n) = isprime(n) && issquare(n-1) \\ Michael B. Porter, Mar 21 2010

is_A002496(n)=issquare(n-1)&&isprime(n) \\ For "random" numbers in the range 10^10 and beyond, at least 5 times faster than the above. - M. F. Hasler, Oct 14 2014

# Python 3.2 or higher required
from itertools import accumulate
from sympy import isprime
A002496_list = [n+1 for n in accumulate(range(10**5),lambda x,y:x+2*y-1) if isprime(n+1)] # Chai Wah Wu, Sep 23 2014

# Python 2.4 or higher required
from sympy import isprime
A002496_list = list(filter(isprime, (n*n+1 for n in range(10**5)))) # David Radcliffe, Jun 26 2016

There are O(sqrt(n)/log(n)) terms of this sequence up to n. But this is just an upper bound. See the Bateman-Horn or Wolf papers, for example, for the conjectured for what is believed to be the correct density.
a(n) = 1 + A005574(n)^2. - R. J. Mathar, Jul 31 2015
Sum_{n>=1} 1/a(n) = A172168. - Amiram Eldar, Nov 14 2020
a(n+1) = 4*A001912(n)^2 + 1. - Hal M. Switkay, Apr 03 2022

Formula, reference, and comment from Charles R Greathouse IV, Aug 24 2009

Edited by M. F. Hasler, Oct 14 2014

A111925 Numbers of the form a^2 + b^4, with a,b > 0.

Original entry on oeis.org

2, 5, 10, 17, 20, 25, 26, 32, 37, 41, 50, 52, 65, 80, 82, 85, 90, 97, 101, 106, 116, 117, 122, 130, 137, 145, 160, 162, 170, 181, 185, 197, 202, 212, 225, 226, 241, 250, 257, 260, 265, 272, 277, 281, 290, 292, 305, 306, 320, 325, 337, 340, 356, 362, 370, 377

Offset: 1

Stefan Steinerberger, Nov 25 2005

nonn

Subsequence of A000404.
Although there are squares, cubes, fifth powers, ... in this sequence, there are no fourth powers. - Altug Alkan, Apr 09 2016
Also, numbers z such that z^5 = x^2 + y^4 for x, y >= 1. - M. F. Hasler, Apr 16 2018
The Friedlander-Iwaniec theorem states that there are infinitely many prime numbers in this sequence. These primes are in A028916. - Bernard Schott, Mar 09 2019

			25 = 3^2 + 2^4, so 25 is an element of the sequence.

R. J. Mathar, Table of n, a(n) for n = 1..1000
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.
Wikipedia, Friedlander-Iwaniec theorem

Cf. A055394, A022549; complement of A111909; subsequence of A000404.

Cf. A028916 (subsequence of primes).

isA111925 := proc(n)
    local a,b ;
    for a from 1 do
        if a^4 >= n then
            return false;
        end if;
        b := n-a^4 ;
        if issqr(b) then
            return true;
        end if;
    end do:
end proc:
A111925 := proc(n)
    option remember;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA111925(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 22 2013

With[{nn=60},Take[Union[First[#]^2+Last[#]^4&/@Tuples[Range[nn],2]],nn]] (* Harvey P. Dale, Jul 09 2014 *)

list(lim)=my(v=List(),t); lim\=1; for(b=1,sqrtnint(lim-1,4), t=b^4; for(a=1,sqrtint(lim-t), listput(v,t+a^2))); Set(v) \\ Charles R Greathouse IV, Jun 07 2016

is(n)=for(b=1,sqrtnint(n-1,4), if(issquare(n-b^4), return(1))); 0 \\ Charles R Greathouse IV, Jun 07 2016

A002645 Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0.

Original entry on oeis.org

2, 17, 97, 257, 337, 641, 881, 1297, 2417, 2657, 3697, 4177, 4721, 6577, 10657, 12401, 14657, 14897, 15937, 16561, 28817, 38561, 39041, 49297, 54721, 65537, 65617, 66161, 66977, 80177, 83537, 83777, 89041, 105601, 107377, 119617, 121937

Offset: 1

N. J. A. Sloane

The largest known quartan prime is currently the largest known generalized Fermat prime: The 1353265-digit 145310^262144 + 1 = (145310^65536)^4 + 1^4, found by Ricky L Hubbard. - Jens Kruse Andersen, Mar 20 2011

Primes of the form (a^2 + b^2)/2 such that |a^2 - b^2| is a square. - Thomas Ordowski, Feb 22 2017

			a(1) =   2 = 1^4 + 1^4.
a(2) =  17 = 1^4 + 2^4.
a(3) =  97 = 2^4 + 3^4.
a(4) = 257 = 1^4 + 4^4.

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
N. D. Elkies, Primes of the form a^4 + b^4, Mathematical Buds, Ed. H. D. Ruderman Vol. 3 Chap. 3 pp. 22-8 Mu Alpha Theta 1984.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Subsequence of A002313 and of A028916.
Intersection of A004831 and A000040.
Cf. A002646, A000583, A003336, A000290, A256852.

a002645 n = a002645_list !! (n-1)
a002645_list = 2 : (map a000040 $ filter ((> 1) . a256852) [1..])
-- Reinhard Zumkeller, Apr 11 2015

nn = 100000; Sort[Reap[Do[n = a^4 + b^4; If[n <= nn && PrimeQ[n], Sow[n]], {a, nn^(1/4)}, {b, a}]][[2, 1]]]
With[{nn=20},Select[Union[Flatten[Table[x^4+y^4,{x,nn},{y,nn}]]],PrimeQ[ #] && #<=nn^4+1&]] (* Harvey P. Dale, Aug 10 2021 *)

upto(lim)=my(v=List(2),t);forstep(x=1,lim^.25,2,forstep(y=2,(lim-x^4)^.25,2,if(isprime(t=x^4+y^4),listput(v,t))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 05 2011

list(lim)=my(v=List([2]),x4,t); for(x=1,sqrtnint(lim\=1,4), x4=x^4; forstep(y=1+x%2,min(sqrtnint(lim-x4,4), x-1),2, if(isprime(t=x4+y^4), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 20 2017

A000040 INTERSECTION A003336. - Jonathan Vos Post, Sep 23 2006

A256852(A049084(a(n))) > 1 for n > 1. - Reinhard Zumkeller, Apr 11 2015

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Nov 07 2002

A243451 Primes of the form n^2 + 16.

Original entry on oeis.org

17, 41, 97, 137, 241, 457, 641, 857, 977, 1697, 2417, 2617, 3041, 4241, 5641, 6257, 6577, 7937, 8297, 9041, 9817, 11897, 13241, 14177, 14657, 15641, 16657, 22817, 27241, 32057, 36497, 44537, 47977, 48857, 52457, 53377, 60041, 62017, 70241, 75641, 78977, 83537

Offset: 1

Vincenzo Librandi, Jun 05 2014

nonn

Intersection of A241751 and A028916; conjecture: sequence is infinite. - Reinhard Zumkeller, Apr 11 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A122062 (associated n).
Cf. similar sequences listed in A243449.
Cf. A010051, A241751; subsequence of A028916.
Primes of form n^2+b^4, b fixed: A002496 (b=1),  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).

a243451 n = a243451_list !! (n-1)
a243451_list = [x | x <- a241751_list, a010051' x == 1]
-- Reinhard Zumkeller, Apr 11 2015

[a: n in [0..1000] | IsPrime(a) where a is n^2+16];

Select[Table[n^2 + 16, {n, 0, 1000}], PrimeQ]
Select[Range[1,301,2]^2+16,PrimeQ] (* Harvey P. Dale, Nov 05 2015 *)

list(lim)=if(lim<17,return([])); my(v=List(),t); forstep(n=1,sqrtint(lim\1-16),2, if(isprime(t=n^2+16), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Aug 18 2017

A256775 Primes of the form n^2 + 81.

Original entry on oeis.org

97, 181, 277, 337, 757, 1237, 2017, 3217, 4177, 5557, 5857, 6481, 7477, 11317, 13537, 16981, 19681, 21397, 33937, 37717, 48481, 51157, 52981, 59617, 62581, 65617, 80737, 84181, 87697, 96181, 102481, 106357, 111637, 119797, 144481, 149077, 155317, 160081

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

subsequence of A045349.
Cf. A010051, A000290; subsequence of A028916.
Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), 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).

a256775 n = a256775_list !! (n-1)
a256775_list = [x | x <- map (+ 81) a000290_list, a010051' x == 1]

[p: p in PrimesUpTo(200000)| IsSquare(p-81)]; // Vincenzo Librandi, Apr 20 2015

Select[Range[400]^2 + 81, PrimeQ] (* Michael De Vlieger, Apr 19 2015 *)

for(n=1,10^3,if(isprime(p=n^2+81),print1(p,", "))) \\ Derek Orr, Apr 24 2015

cp's OEIS Frontend

A331435 a(n) is the least positive k such that A028916(n) - k^2 is a fourth power.

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

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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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Mathematica

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

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PARI

A002496 Primes of the form k^2 + 1.

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A111925 Numbers of the form a^2 + b^4, with a,b > 0.

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Maple

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PARI

A002645 Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0.

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