A002496 -id:A002496 - cp's OEIS frontend

A256839 Primes of form n^2 + 14641.

14657, 14741, 14897, 15217, 15541, 15797, 15937, 19541, 20117, 22037, 22741, 23857, 25457, 28097, 30517, 31541, 38977, 40241, 42197, 43541, 44917, 47041, 48497, 50741, 57077, 58741, 61297, 64817, 65717, 74177, 77141, 80177, 82241, 87541, 107057, 117041

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

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

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), A256838 (b=10), A256840 (b=12), A256841 (b=13).

a256839 n = a256839_list !! (n-1)
a256839_list = [x | x <- map (+ 14641) a000290_list, a010051' x == 1]

Select[Range[500]^2+14641,PrimeQ] (* Harvey P. Dale, Mar 20 2017 *)

A256840 Primes of form n^2 + 20736.

Original entry on oeis.org

20857, 21577, 21961, 23761, 27961, 28657, 29017, 29761, 30937, 33961, 34897, 37897, 41761, 42937, 49297, 51361, 60337, 62761, 65257, 80761, 83737, 93097, 107761, 111337, 113761, 122497, 132961, 142537, 151057, 164377, 173617, 181537, 188017, 192961, 218761

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

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), A256838 (b=10), A256839 (b=11), A256841 (b=13).

a256840 n = a256840_list !! (n-1)
a256840_list = [x | x <- map (+ 20736) a000290_list, a010051' x == 1]

A256841 Primes of form n^2 + 28561.

Original entry on oeis.org

28597, 28661, 28817, 28961, 29137, 29717, 30161, 30497, 30677, 31477, 32917, 33461, 34337, 34961, 35617, 37397, 38561, 42017, 42961, 47057, 49297, 49877, 51061, 55457, 60961, 62417, 64661, 66977, 70177, 70997, 72661, 74357, 75217, 76961, 78737, 86161, 93077

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

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

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), A256838 (b=10), A256839 (b=11), A256840 (b=12).

a256841 n = a256841_list !! (n-1)
a256841_list = [x | x <- map (+ 28561) a000290_list, a010051' x == 1]

Select[Range[300]^2+28561,PrimeQ] (* Harvey P. Dale, Oct 18 2021 *)

A206709 Number of primes of the form b^2 + 1 for b <= 10^n.

Original entry on oeis.org

5, 19, 112, 841, 6656, 54110, 456362, 3954181, 34900213, 312357934, 2826683630, 25814570672, 237542444180, 2199894223892

Offset: 1

Michel Lagneau, Feb 13 2012

Conjecture: The number of primes of the form b^2 + 1 and less than n is asymptotic to 3*n/(4*log(n)).
Examples:
n = 10^3, a(n) = 112 and 3*10^3/(4*log(10^3)) = 108.573...;
n = 10^4, a(n) = 841 and 3*10^4/(4*log(10^4)) = 814.302...;
n = 10^10, a(n) = 312357934 and 3*10^10/(4*log(10^10)) = 325720861.42...
a(n) = A083844(2*n), but not always! The only known exception to this rule is at n = 1. - Arkadiusz Wesolowski, Jul 21 2012
From Jacques Tramu, Sep 14 2018: (Start)
In the table below, K = 0.686413 and pi(10^n) = A000720(10^n):
.
   n         a(n)   K*pi(10^n)
  ==  ===========  ===========
   1            5            3
   2           19           17
   3          112          115
   4          841          843
   5         6656         6584
   6        54110        53882
   7       456362       456175
   8      3954181      3954737
   9     34900213     34902408
  10    312357934    312353959
  11   2826683630   2826686358
  12  25814570672  25814559712
(End)
For a comparison with the estimate that results from the Hardy and Littlewood Conjecture F, see A331942. - Hugo Pfoertner, Feb 03 2020

			a(2) = 19 because there are 19 primes of the form b^2 + 1 for b less than 10^2: 2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101 and 8837.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 264.

Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 2. p. 8.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002496, A083844, A215047, A331941, A331942.

for n from 1 to 9 do : i:=0:for m from 1 to 10^n do:x:=m^2+1:if type(x,prime)=true then i:=i+1:else fi:od: printf ( "%d %d \n",n,i):od:

1 + Accumulate@ Array[Count[Range[10^(# - 1) + 1, 10^#], ?(PrimeQ[#^2 + 1] &)] &, 7] (* _Michael De Vlieger, Sep 18 2018 *)

a(n)=sum(n=1,10^n,ispseudoprime(n^2+1)) \\ Charles R Greathouse IV, Feb 13 2012

from sympy import isprime
def A206709(n):
    c, b, b2, n10 = 0, 1, 2, 10**n
    while b <= n10:
        if isprime(b2):
            c += 1
        b += 1
        b2 += 2*b - 1
    return c # Chai Wah Wu, Sep 17 2018

a(11)-a(12) from Arkadiusz Wesolowski, Jul 21 2012

a(13)-a(14) from Jinyuan Wang, Feb 24 2020

A035092 Smallest k such that (n^2)*k + 1 is prime.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 3, 2, 1, 6, 3, 4, 1, 8, 1, 12, 4, 30, 1, 2, 3, 24, 1, 18, 1, 2, 4, 12, 2, 16, 12, 2, 3, 6, 1, 4, 13, 6, 1, 10, 2, 12, 6, 2, 6, 4, 8, 6, 9, 6, 9, 28, 1, 4, 1, 10, 3, 6, 4, 46, 4, 4, 3, 4, 1, 4, 3, 22, 6, 10, 2, 4, 1, 2, 7, 22, 3, 6, 4, 6, 3, 10, 1, 4, 3, 2, 4, 6, 1, 10, 4, 2, 1

Offset: 1

Labos Elemer

			a(40) = 1 because in 1600k + 1 at k = 1, 1601 is the smallest prime;
a(61) = 46 because in the 46*46*k + 1 sequence the first prime appears at k = 46; it is 171167.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primes in arithmetic progressions

Analogous case is A034693. See also A005574 and A002496.

Table[k = 1; While[! PrimeQ[k (n^2) + 1], k++]; k, {n, 94}] (* Michael De Vlieger, Dec 17 2016 *)

a(n)=k=1;while(!isprime(k*n^2+1),k++);k
vector(100,n,a(n)) \\ Derek Orr, Oct 01 2014

A056905 Primes of the form k^2 + 5.

Original entry on oeis.org

5, 41, 149, 1301, 2309, 5189, 6089, 9221, 13001, 15881, 26249, 28229, 39209, 41621, 60521, 66569, 86441, 112901, 116969, 138389, 171401, 186629, 207941, 213449, 242069, 254021, 266261, 285161, 304709, 331781, 345749, 352841, 389381, 443561

Offset: 1

Henry Bottomley, Jul 07 2000

Except for a(0), a(n) mod 180 = 41 or 149 since k must be a multiple of 6 without being a multiple of 30 for k^2+5 to be prime.

			a(2)=149 since 12^2 + 5 = 149, which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1800
Eric Weisstein's World of Mathematics, Near-Square Prime

Cf. A002496, A056899, A049423, A005473.

[a: n in [0..700] | IsPrime(a) where a is n^2+5]; // Vincenzo Librandi, Nov 30 2011

Select[Table[n^2+5,{n,0,7000}],PrimeQ] (* Vincenzo Librandi, Nov 30 2011 *)

is(n) = ispseudoprime(n) && issquare(n-5) \\ Felix Fröhlich, May 25 2018

a(n) = 36 * A056906(n) + 5.

A069987 Squarefree numbers of form k^2 + 1.

Original entry on oeis.org

2, 5, 10, 17, 26, 37, 65, 82, 101, 122, 145, 170, 197, 226, 257, 290, 362, 401, 442, 485, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1090, 1157, 1226, 1297, 1370, 1522, 1601, 1765, 1937, 2026, 2117, 2210, 2305, 2402, 2501, 2602, 2705, 2810, 2917, 3026

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 01 2002

nonn

Heath-Brown (following Estermann) shows that, for any e > 0, there are k sqrt(x) + O(x^{7/24 + e}) members of this sequence up to x, for k = Product(1 - 2/p^2) = 0.8948412245... (A335963) where the product is over primes p = 1 mod 4. - Charles R Greathouse IV, Nov 19 2012, corrected by Amiram Eldar, Jul 08 2020

Integers k for which the period of the continued fraction of sqrt(k) is 1. - Michel Marcus, Apr 12 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
T. Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105 (1931), pp. 653-662.
D. R. Heath-Brown, Square-free values of n^2 + 1, arXiv:1010.6217 [math.NT], 2010-2012.
D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica 155 (2012), pp. 1-13.

Cf. A059591, A002496, A124809, A005117, A002522, A335963.

select(numtheory:-issqrfree, [seq(n^2+1, n=1..100)]); # Robert Israel, Feb 09 2016

Select[ Range[10^4], IntegerQ[ Sqrt[ # - 1]] && Union[ Transpose[ FactorInteger[ # ]] [[2]]] [[ -1]] == 1 &]
Select[Range[60]^2+1,SquareFreeQ] (* Harvey P. Dale, Mar 21 2013 *)

for(n=1,100,if(issquarefree(n^2+1),print1(n^2+1,",")))

a(n) = A049533(n)^2 + 1.

Edited and extended by Robert G. Wilson v, Benoit Cloitre and Vladeta Jovovic, May 04 2002

A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

Original entry on oeis.org

4, 6, 150, 180, 240, 270, 420, 570, 1290, 1320, 2310, 2550, 2730, 3360, 3390, 4260, 4650, 5850, 5880, 6360, 6780, 9000, 9240, 9630, 10530, 10890, 11970, 13680, 13830, 14010, 14550, 16230, 16650, 18060, 18120, 18540, 19140, 19380, 21600, 21840

Offset: 1

Labos Elemer, Apr 23 2002

Essentially the same as A129293. - R. J. Mathar, Jun 14 2008
Solutions to the equation: A000005(n^4-1) = 8. - Enrique Pérez Herrero, May 03 2012
Terms > 6 are multiples of 30. Subsequence of A070689. - Zak Seidov, Nov 12 2012
{a(n)-1} is a subsequence of A157468; for n>1, {a(n)^2+2} is a subsequence of A242720. - Vladimir Shevelev, Aug 31 2014

			150 is a term since 149, 151 and 22501 are all primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A000720, A001359, A006512, A014574, A002496, A005574, A070156, A070689, A129293, A157468, A242720.

select(n -> isprime(n-1) and isprime(n+1) and isprime(n^2+1), [seq(2*i,i=1..10000)]); # Robert Israel, Sep 02 2014

Do[s=n; If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[1+s^2], Print[n]], {n, 1, 1000000}]
Select[Range[22000],AllTrue[{#+1,#-1,#^2+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 19 2014 *)

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1); \\ Amiram Eldar, Apr 15 2024

For n>1, a(n)^2 = A242720(pi(a(n)-2)) - 2, where pi(n) is the prime counting function (A000720). - Vladimir Shevelev, Sep 02 2014

A088179 Primes p such that mu(p-1) = 1; that is, p-1 is squarefree and has an even number of prime factors, where mu is the Moebius function.

Original entry on oeis.org

2, 7, 11, 23, 47, 59, 83, 107, 167, 179, 211, 227, 263, 331, 347, 359, 383, 463, 467, 479, 503, 547, 563, 571, 587, 691, 719, 839, 859, 863, 887, 911, 967, 983, 1019, 1123, 1187, 1231, 1283, 1291, 1303, 1307, 1319, 1327, 1367, 1439, 1483, 1487, 1523, 1619, 1723

Offset: 1

N. J. A. Sloane and T. D. Noe, Nov 03 2003

nonn

It is an unsolved problem to determine if this sequence has a positive density in the primes. - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003

Except for the initial element 2, this sequence seems to be exactly those primes the sum of whose nonquadratic, nonprimitive-root residues is congruent to -1(mod p). - Dimitri Papadopoulos, Jan 10 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Function

Cf. A049092 (primes p with mu(p-1)=0), A078330 (primes p with mu(p-1)=-1), A089451 (mu(p-1) for prime p).

Cf. A002496.

[n: n in [2..2000] | IsPrime(n) and MoebiusMu(n-1) eq 1]; // Vincenzo Librandi, Jan 10 2016

filter:= proc(p) isprime(p) and numtheory:-mobius(p-1) = 1 end proc:
select(filter, [2,seq(i,i=3..2000,2)]); # Robert Israel, Feb 03 2016

Select[Prime[Range[400]], MoebiusMu[ #-1]==1&]

lista(nn) = forprime(p=2, nn, if (moebius(p-1) == 1, print1(p, ", "))); \\ Michel Marcus, Jan 10 2016

list(lim)=my(v=List(),last); forsquarefree(k=1,lim\1, if(moebius(k)==1, last=k[1], if(k[2][,2]==[1]~ && k[1]-last==1, listput(v,k[1])))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2018

A199307 Primes of the form 4n^3 + 1.

Original entry on oeis.org

5, 109, 257, 1373, 2917, 4001, 27437, 62501, 157217, 202613, 237277, 296353, 470597, 629857, 665501, 1492993, 1556069, 1898209, 2456501, 2634013, 3217429, 3322337, 4244833, 5038849, 5180117, 6572129, 10512289, 11453153, 12706093

Offset: 1

N. J. A. Sloane, Nov 05 2011

Dirichlet's theorem on primes in arithmetic progressions tells us, for example, that there are infinitely many primes of the form 4n+1. For primes represented by polynomials of degree greater than 1, the Bateman-Horn paper gives a conjecture on the density.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
P. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, 16 (1962), 363-367.
Wikipedia, Bateman-Horn Conjecture

Cf. A115104, A001912, A002496, A005574.

[ a: n in [0..200] | IsPrime(a) where a is 4*n^3+1 ]; // Vincenzo Librandi, Nov 08 2011

Select[Table[4n^3+1,{n,1100}],PrimeQ] (* Vincenzo Librandi, Aug 01 2012 *)

for(n=1,1e3,if(isprime(t=4*n^3+1),print1(t", "))) \\ Charles R Greathouse IV, Nov 21 2011

cp's OEIS Frontend

A256839 Primes of form n^2 + 14641.

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A256840 Primes of form n^2 + 20736.

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A256841 Primes of form n^2 + 28561.

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A206709 Number of primes of the form b^2 + 1 for b <= 10^n.

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A035092 Smallest k such that (n^2)*k + 1 is prime.

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A056905 Primes of the form k^2 + 5.

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A069987 Squarefree numbers of form k^2 + 1.

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