A214452 -id:A214452 - 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

A215048 Number of primes of the form 1 + b^4 for 1 < b < 10^n.

Original entry on oeis.org

3, 17, 110, 789, 6395, 52610, 445868, 3857543, 34057327

Offset: 1

Henryk Dabrowski, Aug 01 2012

nonn

Primes 1 + b^4 are a form of generalized Fermat primes. It is conjectured that a(n) is asymptotic to 0.66974*li(10^n).

			a(1) = 3 because the only generalized Fermat primes F_2(b) where b<10^1 are the primes: 17, 257, 1297.

Daniel Shanks, On Numbers of the Form n^4 + 1, Math. Comput. 15 (1961), 186-189.

Yves Gallot, Status of the smallest base values yielding Generalized Fermat primes
Yves Gallot, How many prime numbers appear in a sequence ?
Yves Gallot, A Problem on the Conjecture Concerning the Distribution of Generalized Fermat Prime numbers (a new method for the search for large primes)
Mersenne Wiki, Table of known GF primes b^n+1 where n (exponent) is at least 8192.

Cf. A214452.

Table[Length[Select[Range[2,10^n-1]^4 + 1, PrimeQ]], {n, 5}] (* T. D. Noe, Aug 02 2012 *)

a(n) = sum(b=1, 10^n/2-1, isprime((2*b)^4+1))

a(n) = A214452(4*n) - 1.

A214454 Number of primes of the form x^8 + 1 less than 10^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 7, 8, 13, 15, 23, 29, 41, 52, 65, 84, 114, 149, 194, 257, 336, 431, 547, 686, 880, 1151, 1478, 1961, 2499, 3258, 4264, 5550, 7231, 9416, 12286, 16031, 20887, 27181, 35466, 46009, 60053, 78107, 102086, 133309, 174369, 227810

Offset: 1

Henryk Dabrowski, Jul 18 2012

nonn

It is conjectured that there are infinitely many primes of the form x^8 + 1 (and thus this sequence never becomes constant), but this has not been proved.

			a(16) = 3 because the only primes of the form x^8 + 1 < 10^16 are the primes: 2, 257, 65537.

Chai Wah Wu, Table of n, a(n) for n = 1..90 (terms 1..72 from Henryk Dabrowski)

Cf. A083844, A214452, A214455.

a(n) = sum(k=1,(10^n-1)^(1/8), isprime(k^8+1))

More terms from Henryk Dabrowski, Jul 28 2012

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))

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))

cp's OEIS Frontend

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

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A215048 Number of primes of the form 1 + b^4 for 1 < b < 10^n.

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A214454 Number of primes of the form x^8 + 1 less than 10^n.

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

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

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