A005529 -id:A005529 - 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

A248874 Primitive prime factors of the cyclotomic polynomial sequence Phi(10,k) in the order in which they occur.

Original entry on oeis.org

11, 61, 5, 41, 521, 101, 191, 331, 1181, 9091, 13421, 19141, 2411, 71, 31, 1531, 61681, 9041, 2251, 152381, 185641, 224071, 211, 5791, 9161, 431, 1021, 271, 53951, 401, 71261, 21821, 251, 4051, 1151041, 259631, 132631, 241, 6781, 1824841, 2031671, 41011

Offset: 1

Robert Price, Mar 14 2015

nonn

Phi(10,k) = k^4 - k^3 + k^2 - k + 1.

Note that except for a(3), all terms end with the digit 1.

Robert Price, Table of n, a(n) for n = 1..1218

Cf. A005529.

prim = {}; Do[prim = Join[prim, Complement[First /@ FactorInteger[Cyclotomic[10, k]], prim]], {k, 1000}]; prim

A256144 Primitive prime factors of the cyclotomic polynomial sequence Phi(9,k) in the order in which they occur.

Original entry on oeis.org

3, 73, 757, 19, 829, 2467, 37, 1063, 262657, 333667, 1772893, 80749, 1609669, 397, 18973, 541, 21061, 109, 1270657, 991, 34327, 523, 29989, 64008001, 85775383, 127, 297613, 7792003, 2017, 4987, 5167, 308933353, 433, 8209, 444979, 14437, 41203, 729027001

Offset: 1

Robert Price, Mar 16 2015

nonn

Phi(9,k) = k^6 + k^3 + 1.

Robert Price, Table of n, a(n) for n = 1..1521

Cf. A005529, A248874.

prim = {}; Do[prim = Join[prim, Complement[First /@ FactorInteger[Cyclotomic[9, k]], prim]], {k, 1000}]; prim

lista(nn) = {vs = []; for (n=1, nn, vp = factor(polcyclo(9,n))[,1]; for (i=1, #vp, if (!vecsearch(vs, vp[i]), print1(vp[i], ", "); vs = vecsort(concat(vs, vp[i]),,8););););} \\ Michel Marcus, Mar 20 2015

A256148 Primitive prime factors of the cyclotomic polynomial sequence Phi(3, k) (or Phi(6, k)) in the order in which they occur.

Original entry on oeis.org

3, 7, 13, 31, 43, 19, 73, 37, 157, 61, 211, 241, 307, 127, 421, 463, 79, 601, 757, 271, 67, 331, 151, 1123, 397, 97, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 181, 2551, 379, 919, 409, 2971, 3307, 163, 3541, 523, 3907, 613, 4423, 4831, 1657, 5113, 751

Offset: 1

Robert Price, Mar 16 2015

Phi(3,k) = k^2 + k + 1 and Phi(6,k) = k^2 - k + 1.
Interesting scatter plot.
The terms correspond to the new primes of A081257 in the order of their appearance for n>1 and when A081257(m)>m. - Bill McEachen, Oct 13 2022

Robert Price, Table of n, a(n) for n = 1..733
Bernhard Helmes, Prime sieving on the polynomial f(n)=n^2+n+1.

Cf. A005529, A248874, A256144, A256145, A256146, A081257.

prim = {}; Do[prim = Join[prim, Complement[First /@ FactorInteger[Cyclotomic[6, k]], prim]], {k, 1000}]; prim

lista(nn) = {vs = []; for (n=1, nn, vp = factor(polcyclo(6,n))[,1]; for (i=1, #vp, if (!vecsearch(vs, vp[i]), print1(vp[i], ", "); vs = vecsort(concat(vs, vp[i]),,8););););} \\ Michel Marcus, Mar 20 2015

A256145 Primitive prime factors of the cyclotomic polynomial sequence Phi(8,k) in the order in which they occur.

Original entry on oeis.org

2, 17, 41, 257, 313, 1297, 1201, 241, 193, 73, 137, 7321, 89, 233, 14281, 937, 1489, 65537, 41761, 113, 929, 3833, 160001, 97241, 3209, 139921, 331777, 11489, 26881, 6481, 614657, 353641, 3361, 409, 1129, 61681, 97, 6113, 1336337, 750313, 98801, 10529, 50857

Offset: 1

Robert Price, Mar 16 2015

nonn

Phi(8,k) = k^4 + 1.

Cf. A005529, A096171, A248874, A256144.

prim = {}; Do[prim = Join[prim, Complement[First /@ FactorInteger[Cyclotomic[8, k]], prim]], {k, 1000}]; prim

lista(nn) = {vs = []; for (n=1, nn, vp = factor(polcyclo(8,n))[,1]; for (i=1, #vp, if (!vecsearch(vs, vp[i]), print1(vp[i], ", "); vs = vecsort(concat(vs, vp[i]),,8););););} \\ Michel Marcus, Mar 20 2015

A256146 Primitive prime factors of the cyclotomic polynomial sequence Phi(7,k) in the order in which they occur.

Original entry on oeis.org

7, 127, 1093, 43, 19531, 55987, 29, 4733, 337, 547, 239, 4649, 45319, 659, 4943, 5229043, 8108731, 1743463, 113, 25646167, 449, 80207, 701, 70841, 71, 32719, 631, 3319, 16968421, 5336717, 28771, 321272407, 368089, 4422461, 88009573, 93997, 917087137, 122921

Offset: 1

Robert Price, Mar 16 2015

nonn

Phi(7,k) = k^6 + k^5 + k^4 + k^3 + k^2 + k + 1.

Robert Price, Table of n, a(n) for n = 1..1511

Cf. A005529, A248874, A256144, A256145.

prim = {}; Do[prim = Join[prim, Complement[First /@ FactorInteger[Cyclotomic[7, k]], prim]], {k, 1000}]; prim

lista(nn) = {vs = []; for (n=1, nn, vp = factor(polcyclo(7,n))[,1]; for (i=1, #vp, if (!vecsearch(vs, vp[i]), print1(vp[i], ", "); vs = vecsort(concat(vs, vp[i]),,8););););} \\ Michel Marcus, Mar 20 2015

A256153 Primitive prime factors of the cyclotomic polynomial sequence Phi(5,k) in the order in which they occur.

Original entry on oeis.org

5, 31, 11, 71, 311, 2801, 151, 61, 41, 271, 3221, 22621, 30941, 3761, 4931, 88741, 2711, 911, 251, 40841, 245411, 292561, 346201, 521, 8641, 4561, 637421, 732541, 837931, 17351, 601, 1801, 39451, 22571, 49831, 101, 4271, 194681, 191, 401, 2625641, 579281

Offset: 1

Robert Price, Mar 16 2015

nonn

Phi(5,k) = k^4 + k^3 + k^2 + k + 1.

All terms end with the digit 1.

Robert Price, Table of n, a(n) for n = 1..1191

Cf. A005529, A248874, A256144, A256145, A256146, A256148.

prim = {}; Do[prim = Join[prim, Complement[First /@ FactorInteger[Cyclotomic[5, k]], prim]], {k, 1000}]; prim

lista(nn) = {vs = []; for (n=1, nn, vp = factor(polcyclo(5,n))[,1]; for (i=1, #vp, if (!vecsearch(vs, vp[i]), print1(vp[i], ", "); vs = vecsort(concat(vs, vp[i]),,8););););} \\ Michel Marcus, Mar 20 2015

A120294 Numerator of determinant of n X n matrix with elements M[j,j] = (i+j)/(i+j-1).

Original entry on oeis.org

2, 5, 1, 17, 13, 37, 1, 1, 41, 101, 61, 29, 1, 197, 113, 257, 1, 1, 181, 401, 1, 97, 53, 577, 313, 677, 73, 157, 421, 1, 1, 1, 109, 89, 613, 1297, 137, 1, 761, 1601

Offset: 1

Alexander Adamchuk, Jul 10 2006

Some a(n) are equal to 1 (for n=3,7,8,13,17,18,21,30,31,32,38..=A002312 Arc-cotangent reducible numbers or non-Stormer numbers). All other a(n) (for n=1,2,4,5,6,9,10,11,14,15,16,19,20,22,23..=A005528 Stormer numbers or arc-cotangent irreducible numbers, largest prime factor of n^2 + 1 is >= 2n.) belong to A005529 - Primitive prime factors of the sequence k^2 + 1 (A002522) in the order that they are found. Matrix M[i,j] = (i+j)/(i+j-1) = 1 + 1/(i+j-1) is a sum of n X n unit matrix and n X n Hilbert Matrix. Denominator of determinant of matrix M[i,j] equals determinant of inverse Hilbert matrix A005249.

Cf. A005249, A002312, A005528, A005529, A002522.

Numerator[Table[Det[Table[(i+j)/(i+j-1),{i,1,n},{j,1,n}]],{n,1,40}]]

a(n) = numerator[Det[Table[(i+j)/(i+j-1),{i,1,n},{j,1,n}]]].

A144861 Primitive prime factors of the sequence 2k^2 - 1 in the order in which they are first found.

Original entry on oeis.org

7, 17, 31, 71, 97, 127, 23, 199, 241, 41, 337, 449, 73, 577, 647, 103, 47, 881, 967, 151, 1151, 1249, 193, 1567, 257, 113, 89, 311, 2311, 79, 2591, 2887, 3041, 457, 3361, 3527, 3697, 4049, 4231, 631, 271, 4801, 4999, 743, 5407, 137, 263, 6271, 6961, 313, 1063

Offset: 2

Bernhard Helmes (pi(AT)devalco.de), Sep 23 2008

nonn

Every Mersenne prime number appears on this sequence. a(n) mod 8 = 1 or 7.

Bernhard Helmes, Prime sieving on the polynomial f(n)=2n^2-1.

Cf. A005529, A144848.

Rest[DeleteDuplicates[#[[1]]&/@(Flatten[FactorInteger/@(2*Range[100]^2-1),1])]] (* Harvey P. Dale, Nov 15 2014 *)

Definition clarified by Harvey P. Dale, Nov 15 2014

cp's OEIS Frontend

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

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A248874 Primitive prime factors of the cyclotomic polynomial sequence Phi(10,k) in the order in which they occur.

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A256144 Primitive prime factors of the cyclotomic polynomial sequence Phi(9,k) in the order in which they occur.

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A256148 Primitive prime factors of the cyclotomic polynomial sequence Phi(3, k) (or Phi(6, k)) in the order in which they occur.

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A256145 Primitive prime factors of the cyclotomic polynomial sequence Phi(8,k) in the order in which they occur.

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A256146 Primitive prime factors of the cyclotomic polynomial sequence Phi(7,k) in the order in which they occur.

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A256153 Primitive prime factors of the cyclotomic polynomial sequence Phi(5,k) in the order in which they occur.

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A120294 Numerator of determinant of n X n matrix with elements M[j,j] = (i+j)/(i+j-1).

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