A083844 -id:A083844 - cp's OEIS frontend

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

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.

A214455 Number of primes of the form x^16 + 1 less than 10^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6, 6, 6, 9, 11, 11, 12, 14, 17, 20, 26, 27, 30, 34, 42, 49, 53, 59, 64, 68, 80, 93, 101, 111, 129, 147, 169, 187, 212, 235, 264, 292, 329, 386, 427, 483, 544, 622

Offset: 1

Henryk Dabrowski, Jul 18 2012

nonn

It is conjectured that there are infinitely many primes of the form x^16 + 1 (and thus this sequence never becomes constant), but this has not been proved.

			a(26) = 2 because the only primes or the form x^16 + 1 < 10^26 are the primes: 2, 65537.

Cf. A083844, A214452, A214454.

a(n) = sum(k=1, (10^n-1)^(1/16), isprime(k^16+1))

A172168 Decimal expansion of Sum 1/q, where q is any prime of the form m^2 + 1.

Original entry on oeis.org

8, 1, 4, 5, 9, 6, 5, 7, 1, 7, 0, 2, 9, 7, 2, 8, 4, 5, 2

Offset: 0

Jonathan Vos Post, Jan 28 2010

The sum is trivially convergent because each term is less than the corresponding term of Sum_{j>=1} 1/(j^2) = (Pi^2)/6.

Eight significant digits of this constant are mentioned in A083844, which gives the number of primes of the form m^2 + 1 < 10^n.

			0.8145965717029728452...

G. L. Honaker Jr. and C. Caldwell, 0.81459657, Prime Curios!.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010, pp. 6-8.

Cf. A002496, A005597.

Sum_{q in {primes of form m^2 + 1}} 1/q = Sum_{j>=1} 1/A002496(j) = 1/2 + 1/5 + 1/17 + 1/37 + 1/101 + ...

Leading zero removed and offset adjusted by R. J. Mathar, Jan 30 2010
Corrected and extended by Robert Gerbicz, Mar 13 2010
Name improved by T. D. Noe, Mar 29 2010

A214956 Number of primes of the form x^32 + 1 less than 10^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8

Offset: 1

Henryk Dabrowski, Jul 30 2012

nonn

It is conjectured that there are infinitely many primes of the form x^32 + 1 (and thus this sequence never becomes constant), but this has not been proved.

			a(55) = 2 because the only primes of the form x^32 + 1 < 10^55 are the primes: 2, 185302018885184100000000000000000000000000000001.

Chai Wah Wu, Table of n, a(n) for n = 1..304 (n = 1..128 from Henryk Dabrowski)

Cf. A006315 (k such that k^32+1 is prime).

Cf. A083844, A214452, A214454, A214455.

a(n) = sum(k=1, (10^n-1)^(1/32), isprime(k^32+1))

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

A356364 Number of primes p of the form k^2 + 1 less than 10^n such that p+2 and 2p+1 are also primes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 10, 18, 43, 86, 185, 449, 1091, 2764, 6978, 17951, 47146, 125507, 337600, 916229, 2504458, 6898908

Offset: 1

Angad Singh, Oct 16 2022

			For n = 5, a(5) = 2 since 5 and 25601 are the only two such primes less than or equal to 10^5.

Cf. A006880, A007508, A083844, A092816, A347934.

seq[nmax_] := Module[{c = 0, pow = 10, s = {}, p}, Do[p = k^2 + 1; If[PrimeQ[p] && PrimeQ[p + 2] && PrimeQ[2*p + 1], c++]; If[p > pow, pow *= 10; AppendTo[s, c]], {k, 1, Floor[10^(nmax/2)] + 1}]; s]; seq[13] (* Amiram Eldar, Oct 16 2022 *)

cp's OEIS Frontend

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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A214455 Number of primes of the form x^16 + 1 less than 10^n.

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A172168 Decimal expansion of Sum 1/q, where q is any prime of the form m^2 + 1.

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A214956 Number of primes of the form x^32 + 1 less than 10^n.

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

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Maple

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A356364 Number of primes p of the form k^2 + 1 less than 10^n such that p+2 and 2p+1 are also primes.

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