A259645 -id:A259645 - cp's OEIS frontend

A005574 Numbers k such that k^2 + 1 is prime.

1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250, 256, 260, 264, 270, 280, 284, 300, 306, 314, 326, 340, 350, 384, 386, 396

Offset: 1

N. J. A. Sloane

Hardy and Littlewood conjectured that the asymptotic number of elements in this sequence not exceeding n is approximately c*sqrt(n)/log(n) for some constant c. - Stefan Steinerberger, Apr 06 2006
Also, nonnegative integers such that a(n)+i is a Gaussian prime. - Maciej Ireneusz Wilczynski, May 30 2011
Apparently Goldbach conjectured that any a > 1 from this sequence can be written as a=b+c where b and c are in this sequence (Lemmermeyer link below). - Jeppe Stig Nielsen, Oct 14 2015
No term > 2 can be both in this sequence and in A001105 because of the Aurifeuillean factorization (2*k^2)^2 + 1 = (2*k^2 - 2*k + 1) * (2*k^2 + 2*k + 1). - Jeppe Stig Nielsen, Aug 04 2019

Harvey Dubner, "Generalized Fermat primes", J. Recreational Math., 18 (1985): 279-280.
R. K. Guy, "Unsolved Problems in Number Theory", 3rd edition, A2.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 15, Thm. 17.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
H. Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
F. Ellermann, Primes of the form (m^2)+1 up to 10^6.
L. Euler, Lettre CXLIX (to Goldbach), 1752.
L. Euler, De numeris primis valde magnis, Novi Commentarii academiae scientiarum Petropolitanae 9 (1764), pp. 99-153. See pp. 123-125.
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).
F. Lemmermeyer, Primes of the form a^2+1, Math Overflow question (2010).
Eric Weisstein's World of Mathematics, Landau's Problems.
Eric Weisstein's World of Mathematics, Power.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002522, A001912, A002496, A062325, A090693, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
Other sequences of the type "Numbers k such that k^2 + i is prime": this sequence (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).
Cf. A010051, A259645, A295405 (characteristic function).

a005574 n = a005574_list !! (n-1)
a005574_list = filter ((== 1) . a010051' . (+ 1) . (^ 2)) [0..]
-- Reinhard Zumkeller, Jul 03 2015

[n: n in [0..400] | IsPrime(n^2+1)]; // Vincenzo Librandi, Nov 18 2010

Select[Range[350], PrimeQ[ #^2 + 1] &] (* Stefan Steinerberger, Apr 06 2006 *)
Join[{1},2Flatten[Position[PrimeQ[Table[x^2+1,{x,2,1000,2}]],True]]]  (* Fred Patrick Doty, Aug 18 2017 *)

isA005574(n) = isprime(n^2+1) \\ Michael B. Porter, Mar 20 2010

for(n=1, 1e3, if(isprime(n^2 + 1), print1(n, ", "))) \\ Altug Alkan, Oct 14 2015

from sympy import isprime; [print(n, end = ', ') for n in range(1, 400) if isprime(n*n+1)] # Ya-Ping Lu, Apr 23 2025

a(n) = A090693(n) - 1.

a(n) = 2*A001912(n-1) for n > 1. - Jeppe Stig Nielsen, Aug 04 2019

A056561 Numbers n such that n^2 + n + 41 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78

Offset: 1

Henry Bottomley, Jun 26 2000

Among first 100000 terms, the only run of 13 subsequent values >39 is 219..231. - Zak Seidov, Jan 28 2009
Number of terms less than 10^n: 1, 10, 86, 581, 4149, 31985, 261081, 2208197, 19132652, ... . - Robert G. Wilson v, Apr 20 2015
Complement of A007634. - Robert Israel, Apr 20 2015

			39 is in the sequence because 39^2+39+41=1601 which is prime but 40 is not because 40^2+40+41=1681=41*41.

P. Hoffman, Archimedes' Revenge, pp. 39-40,Penguin Books 1988.

Zak Seidov, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Euler Prime

Cf. A002837, A005846, A007634, A010051, A202018, A259645.

a056561 n = a056561_list !! (n-1)
a056561_list = filter ((== 1) . a010051' . a202018) [0..]
-- Reinhard Zumkeller, Jul 03 2015

[n: n in [0..80] |IsPrime(n^2 + n + 41)]; // Vincenzo Librandi, Sep 28 2012

select(t -> isprime(t^2+t+41), [$0..100]); # Robert Israel, Apr 20 2015

Select[Range[80], PrimeQ[#^2 + # + 41] &] (* Vincenzo Librandi, Sep 28 2012 *)

is(n)=isprime(n^2+n+41) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = (sqrt(4*A005846(n)-163)-1)/2.

a(n) = A002837(n+1)-1. - Robert Price, Nov 08 2019

A087370 Numbers n such that 3n - 1 is a prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 14, 16, 18, 20, 24, 28, 30, 34, 36, 38, 44, 46, 50, 56, 58, 60, 64, 66, 76, 78, 80, 84, 86, 88, 90, 94, 98, 104, 106, 116, 118, 120, 128, 130, 134, 140, 144, 148, 150, 154, 156, 160, 164, 168, 170, 174, 186, 188, 190, 196, 198, 200, 206, 214, 216

Offset: 1

Giovanni Teofilatto, Oct 21 2003

3*n - 1 is an Eisenstein prime. - Vincenzo Librandi, Aug 08 2010

For all elements of this sequence there are no pairs (x,y) of positive integers such that a(n) = 3*x*y - x + y. - Pedro Caceres, Jan 28 2021

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997

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

A003627 gives primes, A091177 gives prime index.

Cf. A010051, subsequence of A016789, A259645.

a087370 n = a087370_list !! (n-1)
a087370_list = filter ((== 1) . a010051' . subtract 1 . (* 3)) [0..]
-- Reinhard Zumkeller, Jul 03 2015

a(n)= A024893(n) + 1 = A088879(n) + 2.

Corrected and extended by Ray Chandler, Oct 22 2003

A342814 Numbers k such that k - 1 and floor(k/5) are both prime.

Original entry on oeis.org

12, 14, 18, 38, 68, 98, 158, 308, 338, 368, 398, 488, 548, 758, 788, 908, 968, 998, 1118, 1568, 1658, 1748, 1868, 1988, 2288, 2438, 2618, 2708, 2858, 2888, 3038, 3068, 3218, 3308, 3458, 3548, 3638, 3698, 3848, 4058

Offset: 1

Claude H. R. Dequatre, Mar 22 2021

Except for a(1) and a(2), all terms == 8 (mod 10).
The first three absolute differences (d) between two consecutive floor(k/5) are respectively equal to 0, 1 and 4 and all the others to 6 or a multiple of 6.
Subsequence of A008864, by definition. - Michel Marcus, Mar 22 2021
For n >= 3, a(n) = 5*A023217(n-2) + 3. Higher terms also coincide with A265767 + 1. - Hugo Pfoertner, Mar 22 2021

			12 is a term because 12 - 1 = 11 and 11 is prime and 12/5 = 2.4 whose floor value is 2 and 2 is also prime.
97 is not a term because 97 - 1 = 96 and 96 is not prime although floor(97/5) = 19 is prime.
Initial terms, associated primes and d:
          k       k - 1     floor(k/5)     d
a(1)     12        11          2
a(2)     14        13          2           0
a(3)     18        17          3           1
a(4)     38        37          7           4
a(5)     68        67         13           6
a(6)     98        97         19           6
a(7)    158       157         31          12
a(8)    308       307         61          30
a(9)    338       337         67           6
a(10)   368       367         73           6

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A007530, A007811, A014561, A259645.

Cf. A008864, A023217, A265767.

R:= NULL:
p:= 1: count:= 0:
while count < 100 do
  p:= nextprime(p);
  if isprime(floor((p+1)/5)) then
     R:= R,p+1; count:= count+1
  fi
od:
R; # Robert Israel, May 22 2024

Select[Range[2,5000,2],And@@PrimeQ[{#-1,Floor[#/5]}]&] (* Giorgos Kalogeropoulos, Apr 01 2021 *)

for(k = 1,10000,if(isprime(k - 1) && isprime(k\5),print1(k", ")))

A342809 Numbers k such that k-1 and round(k/5) are both prime.

Original entry on oeis.org

8, 12, 14, 24, 54, 84, 114, 234, 264, 294, 354, 444, 504, 564, 654, 684, 744, 864, 954, 984, 1164, 1194, 1284, 1554, 1584, 1734, 1914, 2004, 2154, 2214, 2244, 2334, 2394, 2544, 2844, 2964, 3084, 3204, 3414, 3594

Offset: 1

Claude H. R. Dequatre, Mar 22 2021

Except for a(1) and a(2), all terms == 4 (mod 10).
The first three absolute differences (d) between two consecutive rounded (k/5) are respectively equal to 0, 1 and 2 and all the others to 6 or a multiple of 6.
Subsequence of A008864, by definition. - Michel Marcus, Mar 22 2021
For n >= 3, a(n) = 5*A158318(n-2) - 1. - Hugo Pfoertner, Mar 22 2021

			8 is a term because 8 - 1 = 7 and 7 is prime and 8/5 = 1.6 which when rounded gives 2 and 2 is also prime.
235 is not a term because 235 - 1 = 234 and 234 is not a prime although 235/5 = 47 is prime.
Initial terms, associated primes and d:
         k     k - 1   round(k/5)    d
a(1)     8       7         2
a(2)    12      11         2         0
a(3)    14      13         3         1
a(4)    24      23         5         2
a(5)    54      53        11         6
a(6)    84      83        17         6
a(7)   114     113        23         6
a(8)   234     233        47        24
a(9)   264     263        53         6
a(10)  294     293        59         6

Cf. A000040, A007530, A007811, A014561, A259645.

Cf. A008864, A158318.

Select[Range[2,5000,2],And@@PrimeQ[{#-1,Round[#/5]}]&] (* Giorgos Kalogeropoulos, Apr 01 2021 *)

for(k = 1,10000,if(isprime(k - 1) && isprime(k\/5),print1(k", ")))

from sympy import isprime
A342809_list = [k for k in range(1,10**5) if isprime(k-1) and isprime(k//5+int(k % 5 > 2))] # Chai Wah Wu, Apr 08 2021

cp's OEIS Frontend

A005574 Numbers k such that k^2 + 1 is prime.

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A056561 Numbers n such that n^2 + n + 41 is prime.

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A087370 Numbers n such that 3n - 1 is a prime.

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A342814 Numbers k such that k - 1 and floor(k/5) are both prime.

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Maple

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PARI

A342809 Numbers k such that k-1 and round(k/5) are both prime.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python