A002383 -id:A002383 - cp's OEIS frontend

A230252 Number of ways to write n = x + y (x, y > 0) with 2*x + 1, x^2 + x + 1 and y^2 + y + 1 all prime.

0, 1, 2, 3, 2, 3, 4, 4, 4, 3, 4, 1, 3, 3, 3, 5, 5, 4, 3, 6, 4, 7, 7, 2, 4, 6, 4, 4, 6, 3, 1, 4, 2, 4, 7, 4, 1, 4, 4, 2, 6, 4, 3, 4, 2, 3, 5, 3, 2, 1, 2, 3, 6, 2, 6, 6, 3, 5, 4, 5, 3, 7, 2, 4, 6, 2, 4, 5, 3, 5, 8, 5, 2, 10, 4, 4, 8, 5, 6, 7, 8, 4, 11, 4, 3, 6, 4, 2, 4, 8, 8, 11, 5, 3, 11, 5, 3, 6, 4, 5

Offset: 1

Zhi-Wei Sun, Oct 13 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) Any integer n > 3 can be written as p + q with p, 2*p - 3 and q^2 + q + 1 all prime. Also, each integer n > 3 not equal to 30 can be expressed as p + q with p, q^2 + q - 1 and q^2 + q + 1 all prime.
(iii) Any integer n > 1 can be written as x + y (x, y > 0) with x^2 + 1 (or 4*x^2+1) and y^2 + y + 1 (or 4*y^2 + 1) both prime.
(iv) Each integer n > 3 can be expressed as p + q (q > 0) with p, 2*p - 3 and 4*q^2 + 1 all prime.
(v) Any even number greater than 4 can be written as p + q with p, q and p^2 + 4 (or p^2 - 2) all prime. Also, each even number greater than 2 and not equal to 122 can be expressed as p + q with p, q and (p-1)^2 + 1 all prime.
We have verified the first part for n up to 10^8.

			a(5) = 2 since 5 = 2 + 3 = 3 + 2, and 2*2+1 = 5, 2*3+1 = 7, 2^2+2+1 = 7, 3^2+3+1 = 13 are all prime.
a(31) = 1 since 31 = 14 + 17, and 2*14+1 = 29, 14^2+14+1 = 211 and 17^2+17+1 = 307 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A002383, A002384, A002375, A219864, A227908, A227909, A230230, A230241, A230254.

a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[i^2+i+1]&&PrimeQ[(n-i)^2+n-i+1],1,0],{i,1,n-1}]
Table[a[n],{n,1,100}]

A174969 Composites of form n^2 + n + 1.

Original entry on oeis.org

21, 57, 91, 111, 133, 183, 273, 343, 381, 507, 553, 651, 703, 813, 871, 931, 993, 1057, 1191, 1261, 1333, 1407, 1561, 1641, 1807, 1893, 1981, 2071, 2163, 2257, 2353, 2451, 2653, 2757, 2863, 3081, 3193, 3423, 3661, 3783, 4033, 4161, 4291, 4557, 4693, 4971

Offset: 1

Michel Lagneau, Apr 02 2010

nonn

			n=1 gives 1^2+1+1=3, which is prime and so not a term, and similarly for n=2,3; n=4 gives 21=3*7, which is a(1).

L. Poletti, Le serie dei numeri primi appartenente alle due forme quadratiche (A) n^2+n+1 e (B) n^2+n-1 per l'intervallo compreso entro 121 milioni, e cioè per tutti i valori di n fino a 11000, Atti della Reale Accademia Nazionale dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, s. 6, v. 3 (1929), pages 193-218.

Michel Marcus, Table of n, a(n) for n = 1..10000
C. K. Caldwell, Composite Numbers

Cf. A174967, A002384, A002383, A110284.

with(numtheory):for n from 0 to 200 do:x:=n^2+n+1: if type(x,prime)=false then print (x):else fi:od:

Select[Array[ #^2+#+1&,6!,2],!PrimeQ[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Apr 07 2010 *)

isok(k) = (k>1) && !isprime(k) && issquare(4*k-3); \\ Michel Marcus, Apr 20 2022

Example edited and keyword uned removed by D. S. McNeil, Nov 17 2010

A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

Original entry on oeis.org

1, 6, 8, 12, 14, 15, 20, 21, 24, 27, 33, 38, 50, 54, 57, 62, 66, 69, 75, 77, 78, 80, 90, 99, 105, 110, 111, 117, 119, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 168, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218, 231, 236, 245, 246, 266, 272, 278

Offset: 1

Bernard Schott, Dec 18 2012

nonn

All these numbers are in A002384 but not in A053182.
The generated prime numbers n^2 + n + 1 are in A185632.
All the generated numbers n^2 + n + 1 = 111_n are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A002383, A002384, A053182, A053183, A085104, A125134, A185632.

Select[Range@ 280, And[! PrimeQ@ #, PrimeQ[#^2 + # + 1]] &] (* Michael De Vlieger, Jul 30 2017 *)

isok(n) = ! isprime(n) && isprime(n^2 + n + 1); \\ Michel Marcus, Sep 04 2013

A088503 Numbers n such that (n^2 + 3)/4 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 25, 29, 31, 35, 41, 43, 49, 55, 67, 77, 83, 101, 109, 115, 119, 125, 133, 139, 143, 151, 155, 157, 161, 179, 181, 199, 203, 211, 221, 223, 235, 239, 263, 277, 283, 287, 295, 301, 307, 311, 323, 325, 329, 335, 337, 347, 353, 377, 379, 385

Offset: 1

Pierre CAMI, Nov 13 2003

Under Bunyakovsky's conjecture this sequence is infinite. - Charles R Greathouse IV, Dec 28 2011

			(25*25 + 3)/4 = 157, 157 is prime, 25 is the 7th term of the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002383, A002384, A110284.

[n: n in [1..400 by 2] | IsPrime((n^2 + 3) div 4)]; // Vincenzo Librandi, Oct 06 2012

Select[Range[500], PrimeQ[(#^2 + 3)/4] &] (* Vincenzo Librandi, Oct 06 2012 *)

for(k=2,1e3,if(isprime(k^2+k+1),print1(2*k+1", "))) \\ Charles R Greathouse IV, Dec 28 2011

a(n) = 2*A002384(n) + 1 = sqrt(A110284(n)). - Ray Chandler, Sep 07 2005

Corrected and extended by Ray Chandler, Nov 16 2003

A144851 a(n) = number of distinct prime divisors (taken together) of numbers of the form 2x^2+1 for x<=10^n.

Original entry on oeis.org

8, 76, 760, 7445, 73477, 726948, 7218256, 71801859, 715087632, 7127665635, 71089166879, 709344259821

Offset: 1

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

nonn

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

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

d = 10; l = 0; p = 2; 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

A083520 Primes p such that p-1 is a product of two or more consecutive integers. Or (p-1) is a permutation of m items chosen from n, for some m and n. p-1 = k*(k+1)(k+2)...(k+r) for some k and r, r>0.

Original entry on oeis.org

3, 7, 13, 31, 43, 61, 73, 157, 211, 241, 307, 337, 421, 463, 601, 757, 991, 1123, 1321, 1483, 1723, 2521, 2551, 2731, 2971, 3307, 3361, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 6841, 8011, 8191, 9241, 9901, 10303, 10627, 11131, 12211, 12433

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 05 2003

nonn

			61 is in this sequence as 60 = 3*4*5. 73 is in this sequence as 72 = 8*9.

R. J. Mathar, Table of n, a(n) for n = 1..131

Cf. A083521, A002383.

isA083520 := proc(p)
    local k,r,i,po;
    for k from 1 to floor(sqrt(p)) do
        for r from 1 do
            po := product(k+i,i=0..r) ;
            if po  = p-1 then
                return true;
            elif po > p-1 then
                break;
            end if;
        end do:
    end do:
    false ;
end proc:
n := 1 :
for c from 1 do
    p := ithprime(c) ;
    if isA083520(p) then
        printf("%d %d\n",n,p) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 23 2014

More terms from David Wasserman, Nov 19 2004

A087126 Primes of the form p^k - p^(k-1) + 1 for some prime p and integer k > 1.

Original entry on oeis.org

3, 5, 7, 17, 19, 43, 101, 157, 163, 257, 487, 1459, 2029, 4423, 6163, 14407, 19183, 22651, 23549, 26407, 37057, 39367, 62501, 65537, 77659, 113233, 121453, 143263, 208393, 292141, 342733, 375157, 412807, 527803, 564899, 590593, 697049, 843643

Offset: 1

T. D. Noe, Aug 15 2003

nonn

It is usually the case that, for prime p and k > 1, the first time the totient function phi(n) has value p^k - p^(k-1) is for n = p^k. However, this is not true when p^k - p^(k-1) + 1 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Totient Function
Eric Weisstein's World of Mathematics, Totient Valence Function

Cf. A002383 (primes of the form n^2 + n + 1, which is the same as n^2 - n + 1).

Cf. A019434 (Fermat primes), A003306 (2*3^n + 1 is prime), A056799 (8*9^n + 1 is prime), A056797 (9*10^n + 1 is prime), A087139 (least k such that p^k - p^(k-1) + 1 is prime for p = prime(n)).

lst={}; maxNum=10^6; n=1; While[p=Prime[n]; p^2-p+1

A144850 a(n) = number of distinct prime divisors (taken together) of numbers of the form x^2+x+1 for x<=10^n.

Original entry on oeis.org

8, 74, 734, 7233, 71653, 712026, 7090655, 70686855, 705173825, 7038475146, 70278276834, 701910715473

Offset: 1

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

nonn

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

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

d = 10; l = 0; p = 1; c = {}; a = {}; Do[k = p x^2 + x + 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

A236386 Numbers m such that phi(m) is an oblong number.

Original entry on oeis.org

3, 4, 6, 7, 9, 13, 14, 18, 21, 25, 26, 28, 31, 33, 36, 42, 43, 44, 49, 50, 62, 66, 73, 86, 87, 91, 95, 98, 111, 116, 117, 121, 135, 146, 148, 152, 157, 161, 169, 174, 182, 190, 201, 207, 211, 216, 222, 228, 234, 237, 241, 242, 252, 268, 270, 287, 289, 305

Offset: 1

Joseph L. Pe, Jan 24 2014

An oblong number (A002378) is of the form k*(k+1) where k is a natural number.
From Bernard Schott, Feb 27 2023: (Start)
Subsequence of primes is A002383 because in this case phi(k^2+k+1) = k*(k+1).
Subsequence of oblong numbers is A359847 where k and phi(k) are both oblong numbers. (End)

			phi(13) = 12 = 3*4, an oblong number; so 13 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000010, A002378, A002383, A359847.

Similar, but where phi(m) is: A039770 (square), A039771 (cube), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A360944 (triangular).

filter := m -> issqr(1 + 4*phi(m)) : select(filter, [$(1 .. 700)]); # Bernard Schott, Feb 26 2023

Select[Range[500], IntegerQ@Sqrt[1 + 4*EulerPhi[#]] &] (* Giovanni Resta, Jan 24 2014 *)

isok(m) = my(t=eulerphi(m)); !(t%2) && ispolygonal(t/2, 3); \\ Michel Marcus, Feb 27 2023

from itertools import count, islice
from sympy.ntheory.primetest import is_square
from sympy import totient
def A236386_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<2)+1), count(max(1,startvalue)))
A236386_list = list(islice(A236386_gen(),20)) # Chai Wah Wu, Feb 28 2023

a(16)-a(58) from Giovanni Resta, Jan 24 2014

A237360 Numbers n of the form p^2+p+1 (for prime p) such that n^2+n+1 is also prime.

Original entry on oeis.org

57, 381, 993, 4557, 16257, 32943, 49953, 58323, 109893, 135057, 167691, 214833, 237657, 453603, 503391, 564753, 658533, 678153, 780573, 995007, 1248807, 1516593, 1746363, 2218611, 2400951, 3465183, 3738423, 4340973, 4750221, 5232657, 6118203

Offset: 1

Derek Orr, Feb 06 2014

nonn

			57 = 7^2+7+1 (7 is prime) and 57^2+57+1 = 3307 is also prime. Thus, 57 is a member of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A060800, A002383.

for k from 1 do
    p := ithprime(k) ;
    n := numtheory[cyclotomic](3,p) ;
    pn := numtheory[cyclotomic](3,n) ;
    if isprime( pn) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Feb 07 2014

Select[Table[p^2+p+1,{p,Prime[Range[500]]}],PrimeQ[#^2+#+1]&] (* Harvey P. Dale, Feb 09 2014 *)

s=[]; forprime(p=2, 4000, if(isprime(p^4+2*p^3+4*p^2+3*p+3), s=concat(s, p^2+p+1))); s \\ Colin Barker, Feb 07 2014

import sympy
from sympy import isprime
{print(n**2+n+1) for n in range(10**4) if isprime(n) and isprime((n**2+n+1)**2+(n**2+n+1)+1)}

cp's OEIS Frontend

A230252 Number of ways to write n = x + y (x, y > 0) with 2*x + 1, x^2 + x + 1 and y^2 + y + 1 all prime.

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A174969 Composites of form n^2 + n + 1.

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Maple

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A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

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A088503 Numbers n such that (n^2 + 3)/4 is prime.

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Magma

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A144851 a(n) = number of distinct prime divisors (taken together) of numbers of the form 2x^2+1 for x<=10^n.

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A083520 Primes p such that p-1 is a product of two or more consecutive integers. Or (p-1) is a permutation of m items chosen from n, for some m and n. p-1 = k*(k+1)(k+2)...(k+r) for some k and r, r>0.

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Maple

Extensions

A087126 Primes of the form p^k - p^(k-1) + 1 for some prime p and integer k > 1.

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Mathematica

A144850 a(n) = number of distinct prime divisors (taken together) of numbers of the form x^2+x+1 for x<=10^n.

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