A002383 -id:A002383 - cp's OEIS frontend

A108768 Primes that appear in the sequence p:=x^2+x+1, sieved with a quadratic sieve construction.

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

Offset: 1

Bernhard Helmes (pi(AT)devalco.de), Jun 24 2005

This sequence appears in a website available on web.archive (see Quadratic Sieve Construction link). There is a single appearance of the first term 3, while all other primes appear twice. See A256148 for a version of the sequence consistent with the current version of the website where each prime appears only once. - Ray Chandler, Jul 05 2015

Bernhard Helmes, Prime sieving on the polynomial f(n)=n^2+n+1.
Bernhard Helmes, Quadratic Sieve Construction (web.archive).

Cf. A002061, A002383, A007645, A162471, A256148.

// from Quadratic Sieve Construction link.
liste_max:=10000;
for x from 1 to liste_max do
    liste_x[x]:=x^2+x+1;
    liste_prim[x]:=1;
end_for;
x:=1;
while (x1) then
     print ("Prim ", p, "x = ", x, isprime (p)) ;
     // Aussiebung
     while (stelleRay Chandler, Jul 05 2015

A178545 Primes p such that q = p^2 + p + 1 is an emirp.

Original entry on oeis.org

3, 5, 41, 59, 839, 857, 1811, 1931, 3011, 3221, 3407, 3671, 8387, 8543, 8627, 9719, 9743, 9803, 10781, 11549, 12647, 13469, 13487, 13499, 13613, 13931, 14087, 17477, 17573, 17837, 18089, 18269, 19319, 19403, 19661, 19991, 27191, 27947, 31223, 33311, 34313

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 29 2010

It is conjectured (but still an open problem) that there exist infinitely many primes of the form n^2 + n + 1 = ((2*n+1)^2 + 3)/4.
Landau's 4th problem from (1912, 5th Congress of Mathematicians in Cambridge) conjectures that there are infinitely many primes of the form n^2 + 1 (also Euler 1760; Mirsky 1949).
Hardy and Littlewood proposed a conjecture about the asymptotic number of primes of the form n^2 + 1.
An emirp ("prime" spelled backwards) is a prime whose reversal is a different prime, the reversal of q is denoted by R(q).
It is conjectured but also unproved that there are infinitely many emirps (see A048054).
For p > 3 necessarily p of the form 6*k + 5 as (6*k+1)^2 + (6*k+1) + 1 a multiple of 3.

			3^2 + 3 + 1 = 13 = prime(6), R(13) = prime(11), 3 is first term.
5^2 + 5 + 1 = 31 = prime(11), R(31) = prime(6), 5 is 2nd term.
q = 1811^2 + 1811 + 1 = 3281533 = prime(235691), R(q) = prime(240351), first case that p = 1811 = prime(280) = emirp(87) is itself an emirp.

M. Gardner: Die magischen Zahlen des Dr. Matrix, Krueger Verlag, Frankfurt am Main, 1987
R. Guy: Unsolved Problems in Number Theory,3rd edition, Springer, New York, 2004
G. H. Hardy, E. M. Wright: Einfuehrung in die Zahlentheorie, R. Oldenburg, Muenchen, 1958

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A002383, A048054, A006567, A109308, A109309.

filter:= proc(p) local q,qr;
   if not isprime(p) then return false fi;
   q:= p^2+p+1;
   if not isprime(q) then return false fi;
   qr:= revdigs(q);
   qr <> q and isprime(qr);
end proc:
select(filter, [3,seq(i,i=5..50000,6)]); # Robert Israel, Dec 04 2016

EmirpQ[n_] := If[ PrimeQ@n, Block[{id = IntegerDigits@n}, rid = Reverse@ id; rid != id && PrimeQ@ FromDigits@ rid]]; Select[ Prime@ Range@ 3700, EmirpQ[ #^2 + # + 1] &] (* Robert G. Wilson v, Jul 26 2010 *)
p2emrpQ[p_]:=With[{q=p^2+p+1},!PalindromeQ[q]&&AllTrue[{q,IntegerReverse[q]},PrimeQ]]; Select[Prime[Range[3700]],p2emrpQ] (* Harvey P. Dale, Mar 10 2025 *)

More terms from Robert G. Wilson v, Jul 26 2010

A212738 a(n) = (7^p - 6^p - 1)/(1806p) where p is the n-th prime.

Original entry on oeis.org

1, 43, 81271, 3570505, 7025726485, 314435374639, 639872336584027, 60775577624897675065, 2794429652350970000851, 276858360603194024261113585, 600808083611945729624598396925, 28083738921571587634894783049047, 61728002094732427074308383210511683

Offset: 3

Michel Lagneau, May 27 2012

nonn

7^p - 6^p - 1 is divisible by 1806p = 6*7*43*p where p prime > 3 (see the proof with the general case).
The sequence is generalizable with the form a(n) = ((k^p - (k-1)^p - 1)) /(k*(k-1)*p*q) where p = prime(n), k integer such that q = k*(k-1) + 1 prime (q = A002383(n) with k = A055494(n)).
k*(k-1)*p*q divides k^p - (k-1)^p - 1, proof :
(1)   p divides k^p - (k-1)^p - 1 (Fermat’s theorem)
(2)   k*(k-1) divides k^p - (k-1)^p - 1
(3)   q = k*(k-1) + 1 divides k^p - (k-1)^p - 1. Suppose  k^p - (k-1)^p - 1 ==r (mod q). Then ((k-1)^p)*k^p - ((k-1)^p)*(k-1)^p - (k-1)^p ==r*(k-1)^p (mod q). But the first term is congruent to -1 (mod q), the second term is congruent to k^p (mod q) and the last term is congruent to (k-1)^p (mod q). We obtain r (mod q) = r*(k-1)^p (mod q) => r = 0.

Andrew Howroyd, Table of n, a(n) for n = 3..100
Peter Vandendriessche and Hojoo Lee, Problems in elementary number theory, Problem A43.

Cf. A002383, A055494.

with(numtheory): for n from 3 to 25 do:p:=ithprime(n):x:=(7^p - 6^p - 1)/(1806*p): printf(`%d, `, x):od:

a(n)={my(p=prime(n)); (7^p - 6^p - 1)/(1806*p)} \\ Andrew Howroyd, Feb 25 2018

A256497 Triangle read by rows, sums of 2 distinct nonzero cubes: T(n,k) = (n+1)^3+k^3, 1 <= k <= n.

Original entry on oeis.org

9, 28, 35, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 280, 341, 344, 351, 370, 407, 468, 559, 513, 520, 539, 576, 637, 728, 855, 730, 737, 756, 793, 854, 945, 1072, 1241, 1001, 1008, 1027, 1064, 1125, 1216, 1343, 1512, 1729

Offset: 1

Bob Selcoe, Mar 31 2015

When n=k: T(n,k) = (2n+1)(n^2+n+1). Therefore, T(n,k)/(2n+1) = A002061(n+1).
A002383 is the sequence of all primes of the form T(n,k)/(2n+1), n=k.
When starting at T(n,k) n=k, diagonal sums are n^2*(2n+1)^2. For example, starting at T(4,4) = 189: 189+243+351+513 = 4^2*9^2 = 1296.
Coefficients in T(n,k) are multiples of n+k+1; therefore, coefficients in all diagonals starting at T(n,1) are multiples of n+2.
Let T"(n,k) = T(n,k)/(n+k+1). Then reading T"(n,k) by rows:
i. Row sums are A162147(n). For example, T"(3,k) = [65/5, 72/6, 91/7] = [13,12,13]. 13+12+13 = 38; A162147(3) = 38.
ii. Extend the triangle in A215631 to a symmetric array by reflection about the main diagonal, and let that array be T"215631(n,k). Then the diagonal starting with T"215631(n,1) is row n in T"(n,k). For example, the diagonal starting at T"215631(4,1) = [21,19,19,21]; T"(4,k) = [126/6, 133/7, 152/8, 189/9] = [21,19,19,21].
iii.  Coefficients in T"(n,k) are a permutation of A024612.

			Triangle starts:
n\k   1    2    3    4    5    6    7     8    9   10 ...
1:    9
2:    28  35
3:    65  72   91
4:   126  133  152  189
5:   217  224  243  280  341
6:   344  351  370  407  468  559
7:   513  520  539  576  637  728  855
8:   730  737  756  793  854  945  1072 1241
9:   1001 1008 1027 1064 1125 1216 1343 1512 1729
10:  1332 1339 1358 1395 1456 1547 1674 1843 2060 2331
...
The successive terms are (2^3+1^3), (3^3+1^3), (3^3+2^3), (4^3+1^3), (4^3+2^3), (4^3+3^3), ...

Cf. A024670.

Related: A002061, A002383, A003215, A024612, A162147, A215631.

T(n,k) = (n+1)^3+k^3.

T(n,k) = (2k+1)(k^2+k+1) + Sum_{j=k+1..n} A003215(j), n>=k+1. For example, T(8,4) = 9*21 + 91 + 127 + 169 + 217 = 793.

A259631 Numbers k such that the Phi_3(10^10000+k) is prime, where Phi is a cyclic polynomial.

Original entry on oeis.org

8929, 45937, 49256, 50060, 76204, 76855, 125708, 127919, 137050, 137335, 137944, 147466, 163822, 193939, 267131, 295882, 299977, 312610, 322255, 322499, 322988, 370763, 403085, 436060, 458119, 571253, 574597, 601558, 610697, 626978, 627820, 630109, 647039

Offset: 1

Robert Price, Aug 05 2015

nonn

a(53) > 10^6.

Robert Price, Table of n, a(n) for n = 1..52

Cf. A002061, A002383, A002384.

Select[Range[1, 10^6], PrimeQ[(10^10000 + #)^2 + (10^10000 + #) + 1] &]

is(k)=ispseudoprime(subst('x^2+'x+1,'x,10^10000+k)) \\ Charles R Greathouse IV, Aug 05 2015

ABC2 (10^10000+$a)^2 + (10^10000+$a) + 1
a: from 1 to 10000
Charles R Greathouse IV, Aug 05 2015

A319228 Number of primes of the form b^2 + b + 1 for b <= 10^n.

Original entry on oeis.org

6, 32, 189, 1410, 10751, 88118, 745582, 6456835, 56988601, 510007598, 4615215645

Offset: 1

Seiichi Manyama, Sep 14 2018

			a(1) = 6 because there are 6 primes of the form b^2 + b + 1 for b <= 10: 3, 7, 13, 31, 43 and 73.

Cf. A002383, A002384, A206709.

{a(n) = sum(k=0, 10^n, isprime(k^2+k+1))}

from sympy import isprime
def A319228(n):
    c, b, b2, n10 = 0, 1, 3, 10**n
    while b <= n10:
        if isprime(b2):
            c += 1
        b += 1
        b2 += 2*b
    return c # Chai Wah Wu, Sep 17 2018

a(10) from Chai Wah Wu, Sep 17 2018

a(11) from Chai Wah Wu, Sep 18 2018

A343775 Primes that are neither of the form (c^q+1)/(c+1) and nor of the form (b^k-1)/(b-1) for any b, c > 1 and k, q primes > 2.

Original entry on oeis.org

2, 5, 17, 19, 23, 29, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367

Offset: 1

Bernard Schott, Apr 29 2021

Equivalently, non-Brazilian primes that are not of the form (c^q+1)/(c+1) for some c > 1, q prime > 2.

Equals A220627 \ A059055.

Primes of the form (b^k-1)/(b-1) = A085104 (Brazilian primes).
Primes of the form (c^q+1)/(c+1) = A059055.
Primes of the form (b^k-1)/(b-1) and also (c^q+1)/(c+1): A002383 \ {3} is a subsequence, but, maybe the intersection (conjecture).
Primes of the form (b^k-1)/(b-1) but not (c^q+1)/(c+1) = A225148.
Primes of the form (c^q+1)/(c+1) but not (b^k-1)/(b-1) = A343774.
Primes neither of the form (c^q+1)/(c+1) nor (b^k-1)/(b-1) = this sequence.

isc(p) = for (b=2, p, my(k=3); while ((x=(b^k+1)/(b+1)) <= p, if (x == p, return (1)); k = nextprime(k+1); ); );
isnotb(p) = for (b=2, p-1, my(d=digits(p, b), md=vecmin(d)); if ((#d > 2) && (md == 1) && (vecmax(d) == 1), return (0)); ); return (1);
isok(p) = isprime(p) && !isc(p) && isnotb(p); \\ Michel Marcus, May 01 2021

A353056 Composite numbers of the form k^2+k+1 all of whose prime factors are of that same form.

Original entry on oeis.org

21, 91, 273, 343, 507, 651, 1333, 4557, 6321, 6643, 27391, 36673, 50851, 65793, 83811, 105301, 139503, 190533, 194923, 217623, 234741, 391251, 545383, 1647373, 1961401, 2032051, 2376223, 4517751, 6118203, 6484663, 11590621, 13180531, 14535157, 20155611, 28371603, 35646871

Offset: 1

Michel Marcus, Apr 20 2022

nonn

			21 = 4^2+4+1 and its factors are 3 and 7, terms of A002383. So 21 is a term.

David A. Corneth, Table of n, a(n) for n = 1..318
Cody S. Hansen and Pace P. Nielsen, Prime factors of phi3(x) of the same form, arXiv:2204.08971 [math.NT], 2022.

Subsequence of A174969.

Cf. A002383.

q:= n-> not isprime(n) and andmap(p-> issqr(4*p-3), numtheory[factorset](n)):
select(q, [k*(k+1)+1$k=4..6000])[];  # Alois P. Heinz, Apr 20 2022

Select[Table[n^2 + n + 1, {n, 1, 6000}], CompositeQ[#] && AllTrue[FactorInteger[#][[;; , 1]], IntegerQ@Sqrt[4*#1 - 3] &] &] (* Amiram Eldar, Apr 20 2022 *)

lista(nn) = {for (n=1, nn, my(x=n^2+n+1); if (! isprime(x), my(fa=factor(x), ok=1); for (k=1, #fa~, my(fk=fa[k,1]); if (! issquare(4*fk-3), ok = 0);); if (ok, print1(x, ", "));););}

from sympy import isprime, factorint
from itertools import count, takewhile
def agento(N): # generator of terms up to limit N
    form = set(takewhile(lambda x: x<=N, (k**2 + k + 1 for k in count(1))))
    for t in sorted(form):
        if not isprime(t) and all(p in form for p in factorint(t)):
            yield t
print(list(agento(10**8))) # Michael S. Branicky, Apr 20 2022

A108154 a(n) = n^2 - n + 1 if prime else 0.

Original entry on oeis.org

0, 3, 7, 13, 0, 31, 43, 0, 73, 0, 0, 0, 157, 0, 211, 241, 0, 307, 0, 0, 421, 463, 0, 0, 601, 0, 0, 757, 0, 0, 0, 0, 0, 1123, 0, 0, 0, 0, 1483, 0, 0, 1723, 0, 0, 0, 0, 0, 0, 0, 0, 2551, 0, 0, 0, 2971, 0, 0, 3307, 0, 3541, 0, 0, 3907, 0, 0, 0, 4423, 0, 0, 4831, 0, 5113, 0, 0, 0, 5701, 0

Offset: 1

Pierre CAMI, Jun 06 2005

			1^2 - 1 + 1 = 1, which is not prime, so a(1)=0;
2^2 - 2 + 1 = 3, which is prime, so a(2)=3;
3^3 - 3 + 1 = 7, which is prime, so a(3)=7.

Cf. A002061, A002383, A002384.

f[n_] := Block[{p = n^2 - n + 1}, If[ PrimeQ[p], p, 0]]; Table[ f[n], {n, 77}] (* Robert G. Wilson v, Jun 07 2005 *)

vector(80, n, if (isprime(p=n^2-n+1), p, 0)) \\ Michel Marcus, Jul 31 2015

More terms from Robert G. Wilson v, Jun 07 2005

A144852 a(n) = number of distinct prime divisors (taken together) of numbers of the form 4x^2+1 for x<=10^n.

Original entry on oeis.org

9, 87, 836, 8000, 78124, 766585, 7556731, 74771106, 741554656, 7366252759, 73261462211, 729280694469

Offset: 1

Artur Jasinski & Bernhard Helmes (bhelmes(AT)gmx.de), Sep 22 2008

nonn

Primes of the form 4x^2+1 see A121326(n) = A002496(n+1).

Bernhard Helmes, Prime sieving on the polynomial f(n)=4n^2+1.

Cf. A002383, A121326, A143835, A143868, A144848, A144850, A144851.

d = 10; l = 0; p = 4; c = {}; a = {}; Do[k = p x^2 + 1; b = Divisors[k]; Do[If[PrimeQ[b[[n]]], AppendTo[a, b[[n]]]], {n, 1, Length[b]}]; If[x == d, a = Union[a]; l = Length[a]; d = 10 d; Print[l]; AppendTo[c, l]], {x, 1, 10000}]; c (*Artur Jasinski*)

Fixed broken link, corrected and extended to agree with website. - Ray Chandler, Jun 30 2015

cp's OEIS Frontend

A108768 Primes that appear in the sequence p:=x^2+x+1, sieved with a quadratic sieve construction.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

MuPAD

A178545 Primes p such that q = p^2 + p + 1 is an emirp.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A212738 a(n) = (7^p - 6^p - 1)/(1806p) where p is the n-th prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

A256497 Triangle read by rows, sums of 2 distinct nonzero cubes: T(n,k) = (n+1)^3+k^3, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A259631 Numbers k such that the Phi_3(10^10000+k) is prime, where Phi is a cyclic polynomial.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PFGW

A319228 Number of primes of the form b^2 + b + 1 for b <= 10^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Python

Extensions

A343775 Primes that are neither of the form (c^q+1)/(c+1) and nor of the form (b^k-1)/(b-1) for any b, c > 1 and k, q primes > 2.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A353056 Composite numbers of the form k^2+k+1 all of whose prime factors are of that same form.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica