A083849 -id:A083849 - cp's OEIS frontend

A083844 Number of primes of the form x^2 + 1 < 10^n.

2, 4, 10, 19, 51, 112, 316, 841, 2378, 6656, 18822, 54110, 156081, 456362, 1339875, 3954181, 11726896, 34900213, 104248948, 312357934, 938457801, 2826683630, 8533327397, 25814570672, 78239402726, 237542444180, 722354138859, 2199894223892

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that there are infinitely many primes of the form x^2 + 1 (and thus this sequence never becomes constant), but this has not been proved.

These primes can be found quickly using a sieve based on the fact that numbers of this form have at most one primitive prime factor (A005529). The sum of the reciprocals of these primes is 0.81459657... - T. D. Noe, Oct 14 2003

			a(3) = 10 because the only primes or the form x^2 + 1 < 10^3 are the ten primes: 2, 5, 17, 37, 101, 197, 257, 401, 577, 677.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

C. K. Caldwell, An Amazing Prime Heuristic A pdf file.
Jon Grantham, Parallel Computation of Primes of the form x^2+1
Jon Grantham and Hester Graves, Primes of the Form m^2+1 and Goldbach's 'Other Other' Conjecture, arXiv:2502.03513 [math.NT], 2025.
Apoloniusz Tyszka, 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.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A005574, A002496, A083845, A083846, A083847, A083848, A083849, A214452, A214454, A214455.

Cf. A005529 (primitive prime factors of the sequence k^2+1).

c = 1; k = 2; (* except for the initial prime 2, all X's must be odd. *) Do[ While[ k^2 + 1 < 10^n, If[ PrimeQ[k^2 + 1], c++ ]; k += 2]; Print[c], {n, 1, 20}]

Edited by Robert G. Wilson v, May 08 2003
More terms from T. D. Noe, Oct 14 2003
a(17)-a(22) from Robert Gerbicz, Apr 15 2009
a(23)-a(25) from Marek Wolf and Robert Gerbicz (code from Robert, computation done by Marek) Robert Gerbicz, Mar 13 2010
a(26)-a(28) from Jon Grantham, Jan 18 2017
a(28) corrected by Jon Grantham, Jan 30 2018

A083847 a(n) = number of primes of the form x^2 + 1 <= 2^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 24, 33, 42, 54, 70, 91, 114, 158, 212, 293, 393, 539, 713, 957, 1301, 1792, 2459, 3378, 4615, 6233, 8418, 11540, 15867, 21729, 29843, 41169, 56534, 77697, 106787, 147067, 203025, 280340, 387308, 535153, 739671, 1023655, 1416635, 1960813, 2716922, 3764693, 5218926, 7238715

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence does not become a constant, but this has never been proved.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

C. K. Caldwell, An amazing prime heuristic.
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A005574, A002496, A083844 - A083849.

a(n) = my(nb = 0); forprime(p=2, 2^n, if (issquare(p-1), nb++);); nb  \\ Michel Marcus, Jun 14 2013

More terms from Alexander D. Healy, Feb 06 2005

A083845 a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 10^n.

Original entry on oeis.org

2, 6, 26, 94, 314, 986, 3160, 9990, 31614, 99996, 316206, 999960, 3162246, 9999960, 31622764, 99999966, 316227734, 999999924, 3162277654, 9999999956, 31622776500, 99999999964, 316227766006, 999999999886, 3162277660140

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence never becomes a constant, but this has not been proved.

The ratio a(n+2)/a(n) appears to approach 10, as one might expect. - Bill McEachen, Nov 03 2013

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A005574, A002496, A083844, A083846, A083847, A083848, A083849.

Do[ k = Floor[ Sqrt[ 10^n] - 1]; While[ !PrimeQ[k^2 + 1], k-- ]; Print[k], {n, 1, 25}]

Edited and extended by Robert G. Wilson v, May 08 2003

A083846 a(n) is the largest prime of the form x^2 + 1 <= 10^n.

Original entry on oeis.org

5, 37, 677, 8837, 98597, 972197, 9985601, 99800101, 999444997, 9999200017, 99986234437, 999920001601, 9999799764517, 99999200001601, 999999202999697, 9999993200001157, 99999979750774757, 999999848000005777

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence does not become a constant, but this has not been proved. It is easily shown that all terms greater than 5 end in 1 or 7.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A005574, A002496, A083844, A083845, A083847, A083848, A083849.

Do[ k = Floor[ Sqrt[ 10^n] - 1]; While[ !PrimeQ[k^2 + 1], k-- ]; Print[k^2 + 1], {n, 1, 19}]
lpf[n_]:=Module[{p=NextPrime[10^n,-1]},While[!IntegerQ[Sqrt[p-1]],p= NextPrime[ p,-1]];p]; Array[lpf,10] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Feb 11 2023 *)

Edited and extended by Robert G. Wilson v, May 08 2003

A083848 a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 2^n.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 10, 14, 20, 26, 40, 56, 90, 126, 180, 250, 350, 496, 716, 1010, 1440, 2034, 2896, 4086, 5774, 8184, 11566, 16380, 23166, 32766, 46326, 65534, 92666, 131070, 185354, 262130, 370714, 524260, 741454, 1048554, 1482904, 2097146

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that this sequence is infinite, but this has never been proved.

Ratio of successive terms appears to approach sqrt(2). - Bill McEachen, Nov 03 2013

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A005574, A002496, A083844, A083845, A083846, A083847, A083849.

A174246 Number of primes of the form x^2 + 1 < 2^n.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 24, 33, 42, 54, 70, 91, 114, 158, 212, 293, 393, 539, 713, 957, 1301, 1792, 2459, 3378, 4615, 6233, 8418, 11540, 15867, 21729, 29843, 41169, 56534, 77697, 106787, 147067, 203025, 280340, 387308, 535153, 739671

Offset: 1

Robert Gerbicz, Mar 13 2010

nonn

Terms from Marek Wolf and Robert Gerbicz (code from Robert, computation done by Marek).
It is conjectured that this sequence is unbounded, but this has never been proved. [Comment corrected by Kellen Myers, Oct 12 2014.]
More precisely, it is not known if there are infinitely many primes of the form k^2 + 1. See references and links. - N. J. A. Sloane, Oct 14 2014
Same as A083847 except for a(1) = 0. - Georg Fischer, Oct 14 2018

			a(10) = 10 because the only primes or the form x^2 + 1 < 2^10 are the ten primes: 2, 5, 17, 37, 101, 197, 257, 401, 577 & 677.

Chris Caldwell, Prime Conjectures and Open Questions
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
Eric W. Weisstein, Landau's Problems

Cf. A083844, A083845, A083846, A083847, A083848, A083849, A002496.

N:= 30: # to get a(1) to a(N).
P:= select(isprime,[2,seq((2*i)^2+1, i = 1 .. floor(sqrt(2^N-1)/2))]):
seq(nops(select(`<`,P,2^n)), n=1..N); # Robert Israel, Oct 13 2014

lista(nn) = {nb = 0; for (n=1, nn, forprime(p=2^n, 2^(n+1)-1, if (issquare(p-1), nb++);); print1(nb, ", "););} \\ Michel Marcus, Oct 13 2014

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A083844 Number of primes of the form x^2 + 1 < 10^n.

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A083847 a(n) = number of primes of the form x^2 + 1 <= 2^n.

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A083845 a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 10^n.

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A083846 a(n) is the largest prime of the form x^2 + 1 <= 10^n.

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A083848 a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 2^n.

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A174246 Number of primes of the form x^2 + 1 < 2^n.

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