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 1-10 of 13 results. Next

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

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Author

Keywords

Comments

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]

References

  • 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.

Crossrefs

Cf. A083844 (number of these primes < 10^n), A199401 (growth constant).
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).

Programs

  • Haskell
    a002496 n = a002496_list !! (n-1)
    a002496_list = filter ((== 1) . a010051') a002522_list
    -- Reinhard Zumkeller, May 06 2013
    
  • Magma
    [p: p in PrimesUpTo(100000)| IsSquare(p-1)]; // Vincenzo Librandi, Apr 09 2011
    
  • Maple
    select(isprime, [2, seq(4*i^2+1, i= 1..1000)]); # Robert Israel, Oct 14 2014
  • Mathematica
    Select[Range[100]^2+1, PrimeQ]
    Join[{2},Select[Range[2,300,2]^2+1,PrimeQ]] (* Harvey P. Dale, Dec 18 2018 *)
  • PARI
    isA002496(n) = isprime(n) && issquare(n-1) \\ Michael B. Porter, Mar 21 2010
    
  • PARI
    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
    # 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
    # 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

Formula

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

Extensions

Formula, reference, and comment from Charles R Greathouse IV, Aug 24 2009
Edited by M. F. Hasler, Oct 14 2014

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

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Author

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.
		

Crossrefs

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

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

Views

Author

Vincenzo Librandi, Jun 05 2014

Keywords

Comments

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

Crossrefs

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).

Programs

  • Haskell
    a243451 n = a243451_list !! (n-1)
    a243451_list = [x | x <- a241751_list, a010051' x == 1]
    -- Reinhard Zumkeller, Apr 11 2015
    
  • Magma
    [a: n in [0..1000] | IsPrime(a) where a is n^2+16];
    
  • Mathematica
    Select[Table[n^2 + 16, {n, 0, 1000}], PrimeQ]
    Select[Range[1,301,2]^2+16,PrimeQ] (* Harvey P. Dale, Nov 05 2015 *)
  • PARI
    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

Views

Author

Reinhard Zumkeller, Apr 11 2015

Keywords

Comments

Conjecture: sequence is infinite.

Crossrefs

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).

Programs

  • Haskell
    a256775 n = a256775_list !! (n-1)
    a256775_list = [x | x <- map (+ 81) a000290_list, a010051' x == 1]
    
  • Magma
    [p: p in PrimesUpTo(200000)| IsSquare(p-81)]; // Vincenzo Librandi, Apr 20 2015
    
  • Mathematica
    Select[Range[400]^2 + 81, PrimeQ] (* Michael De Vlieger, Apr 19 2015 *)
  • PARI
    for(n=1,10^3,if(isprime(p=n^2+81),print1(p,", "))) \\ Derek Orr, Apr 24 2015

A256777 Primes of form n^2 + 625.

Original entry on oeis.org

641, 661, 769, 821, 881, 1109, 1201, 1301, 1409, 2069, 2389, 2741, 3329, 3541, 3761, 3989, 4721, 6101, 6709, 7349, 7681, 8369, 9461, 12289, 14081, 14549, 16001, 18049, 19121, 20789, 25589, 28181, 31601, 32309, 33749, 35221, 35969, 37489, 38261, 39041, 39829
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2015

Keywords

Comments

Conjecture: sequence is infinite.

Crossrefs

Cf. A010051, A000290; subsequence of A028916.
Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), 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
    a256777 n = a256777_list !! (n-1)
    a256777_list = [x | x <- map (+ 625) a000290_list, a010051' x == 1]
    
  • Mathematica
    Select[Range[200]^2+625,PrimeQ] (* Harvey P. Dale, Aug 27 2025 *)
  • PARI
    list(lim)=if(lim<641,return([])); my(v=List(),t); forstep(n=4, sqrtint(lim\1-625), 2, if(isprime(t=n^2+625), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Aug 18 2017

A256834 Primes of form n^2 + 1296.

Original entry on oeis.org

1297, 1321, 1657, 2137, 2521, 3697, 5521, 6337, 7537, 8521, 10321, 11497, 13177, 15937, 16921, 18457, 23497, 24097, 25321, 34057, 35521, 40897, 43321, 45817, 47521, 58417, 59377, 88321, 90697, 94321, 98017, 106921, 109537, 117577, 127321, 131617, 138937
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2015

Keywords

Comments

Conjecture: sequence is infinite.

Crossrefs

Cf. A010051, A000290; subsequence of A028916.
Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

Programs

  • Haskell
    a256834 n = a256834_list !! (n-1)
    a256834_list = [x | x <- map (+ 1296) a000290_list, a010051' x == 1]
  • Mathematica
    Select[Range[1,401,2]^2+1296,PrimeQ] (* Harvey P. Dale, Sep 18 2018 *)

A256835 Primes of form n^2 + 2401.

Original entry on oeis.org

2417, 2437, 2657, 2801, 3301, 3557, 3697, 4001, 4337, 4517, 7877, 10501, 11617, 12401, 13217, 19301, 20357, 20897, 26737, 28001, 29297, 33377, 36997, 38501, 40037, 44017, 48197, 49057, 64901, 70001, 77477, 78577, 86501, 90017, 92401, 104801, 107377, 108677
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2015

Keywords

Comments

Conjecture: sequence is infinite.

Crossrefs

Cf. A010051, A000290; subsequence of A028916.
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), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

Programs

  • Haskell
    a256835 n = a256835_list !! (n-1)
    a256835_list = [x | x <- map (+ 2401) a000290_list, a010051' x == 1]

A256836 Primes of form n^2 + 4096.

Original entry on oeis.org

4177, 4217, 4457, 4721, 4937, 6121, 7121, 7577, 7817, 9137, 9721, 10337, 10657, 11321, 12377, 13121, 15121, 16417, 17321, 18257, 23417, 23977, 25121, 26297, 31321, 34721, 36137, 36857, 38321, 40577, 44497, 47777, 50321, 52057, 52937, 54721, 57457, 81937
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2015

Keywords

Comments

Conjecture: sequence is infinite.

Crossrefs

Cf. A010051, A000290; subsequence of A028916.
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), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

Programs

  • Haskell
    a256836 n = a256836_list !! (n-1)
    a256836_list = [x | x <- map (+ 4096) a000290_list, a010051' x == 1]
  • Mathematica
    Select[Range[1,300,2]^2+4096,PrimeQ] (* Harvey P. Dale, May 26 2025 *)

A256837 Primes of form n^2 + 6561.

Original entry on oeis.org

6577, 6661, 6961, 7237, 7717, 8161, 8677, 9697, 10657, 12037, 16561, 17377, 18661, 21937, 24517, 25057, 26161, 33457, 35461, 37537, 56737, 57637, 69061, 74161, 77317, 81637, 84961, 106417, 108961, 124897, 129061, 143461, 146437, 147937, 150961, 155557
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2015

Keywords

Comments

Conjecture: sequence is infinite.

Crossrefs

Cf. A010051, A000290; subsequence of A028916.
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), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13)

Programs

  • Haskell
    a256837 n = a256837_list !! (n-1)
    a256837_list = [x | x <- map (+ 6561) a000290_list, a010051' x == 1]

A256838 Primes of form n^2 + 10000.

Original entry on oeis.org

10009, 10169, 10289, 10529, 10729, 11369, 11681, 12401, 12601, 12809, 13249, 13721, 14489, 15329, 16561, 16889, 17569, 17921, 19801, 20201, 21881, 22769, 23689, 26641, 27689, 29881, 30449, 32801, 33409, 34649, 35281, 37889, 38561, 39241, 39929, 48809, 53681
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2015

Keywords

Comments

Conjecture: sequence is infinite.

Crossrefs

Cf. A010051, A000290; subsequence of A028916.
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), A256839 (b=11), A256840 (b=12), A256841 (b=13).

Programs

  • Haskell
    a256838 n = a256838_list !! (n-1)
    a256838_list = [x | x <- map (+ 10000) a000290_list, a010051' x == 1]
Showing 1-10 of 13 results. Next