A002384 -id:A002384 - cp's OEIS frontend

A089358 Numbers k such that k^2 - 3*k + 3 is prime.

3, 4, 5, 7, 8, 10, 14, 16, 17, 19, 22, 23, 26, 29, 35, 40, 43, 52, 56, 59, 61, 64, 68, 71, 73, 77, 79, 80, 82, 91, 92, 101, 103, 107, 112, 113, 119, 121, 133, 140, 143, 145, 149, 152, 155, 157, 163, 164, 166, 169, 170, 175, 178, 190, 191, 194, 196, 205, 208

Offset: 1

Giovanni Teofilatto, Dec 27 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Select[Range[210],PrimeQ[#^2-3#+3]&] (* Harvey P. Dale, Mar 23 2012 *)

is(n)=isprime(n^2-3*n+3) \\ Charles R Greathouse IV, Jun 12 2017

a(n) = A002384(n)+2.

Corrected by Harvey P. Dale, Mar 23 2012

A117307 Numbers for which (phi(n))^2+phi(n)+1 is a prime number.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 13, 14, 15, 16, 18, 20, 21, 24, 25, 26, 28, 30, 33, 35, 36, 39, 42, 44, 45, 50, 52, 56, 66, 67, 70, 72, 78, 79, 81, 84, 90, 121, 123, 134, 139, 151, 158, 162, 163, 164, 165, 176, 193, 200, 203, 215, 220, 221, 242, 243, 245, 246, 249

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 24 2006

nonn

			14 is in the sequence because (phi(14))^2+phi(14)+1 = 6^2+6+1 = 43, which is a prime number.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A002384.

f[x_] := x^2 + x + 1; Select[Range[250], PrimeQ[f[EulerPhi[#]]] &] (* Amiram Eldar, Feb 08 2021 *)

lista(nn) = {for (n = 1, nn, if (isprime((eulerphi(n))^2 + eulerphi(n) + 1), print1(n, ", ")););} \\ Michel Marcus, Jun 01 2013

Corrected by Michel Marcus, Jun 01 2013

A245476 Least number k > 1 such that k^n + k + 1 is prime, or 0 if no such number exists.

Original entry on oeis.org

2, 2, 2, 2, 0, 2, 2, 0, 3, 3, 0, 2, 5, 0, 2, 2, 0, 2, 8, 0, 6, 3, 0, 6, 15, 0, 6, 2, 0, 2, 23, 0, 23, 56, 0, 15, 114, 0, 14, 11, 0, 3, 14, 0, 29, 110, 0, 21, 9, 0, 53, 59, 0, 6, 2, 0, 3, 29, 0, 71, 21, 0, 146, 17, 0, 35, 2, 0, 9, 6, 0, 77, 41, 0, 27, 176, 0, 153, 21, 0, 39, 32, 0, 2, 314, 0, 3, 5, 0, 66, 44, 0, 234

Offset: 1

Derek Orr, Jul 23 2014

nonn

Except for a(2), a(n) = 0 if n == 2 mod 3 (A016789).
It appears that this is an "if and only if".
a(n) = 2 if and only if n is in A057732.
Many terms in the linked table correspond to probable primes. If n == 2 mod 3 then k^2+k+1 divides k^n+k+1. This is why a(n) = 0 if n > 2 and n == 2 mod 3. - Jens Kruse Andersen, Jul 28 2014

			2^9 + 2 + 1 = 515 is not prime. 3^9 + 3 + 1 = 19687 is prime. Thus a(9) = 3.

Robert Israel and Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 (first 640 terms from Robert Israel)

Cf. A127599, A016789, A057732.

Cf. Numbers n such that n^s + n + 1 is prime: A005097 (s = 1), A002384 (s = 2), A049407 (s = 3), A049408 (s = 4), A075723 (s = 6), A075722 (s = 7), A075720 (s = 9), A075719 (s = 10), A075718 (s = 12), A075717 (s = 13), A075716 (s = 15), A075715 (s = 16), A075714 (s = 18), A075713 (s = 19).

f:= proc(n) local k;
   if n mod 3 = 2 and n > 2 then return 0 fi;
   for k from 2 to 10^6 do
      if isprime(k^n+k+1) then return k fi
   od:
  error("no solution found for n = %1",n);
end proc:
seq(f(n),n=1..100); # Robert Israel, Jul 27 2014

a(n) = if(n>2&&n==Mod(2, 3), return(0)); k=2; while(!ispseudoprime(k^n+k+1), k++); k
vector(150, n, a(n)) \\ Derek Orr with corrections and improvements from Colin Barker, Jul 23 2014

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

A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.

Original entry on oeis.org

18, 50, 578, 1458, 3362, 4802, 6962, 10082, 15842, 20402, 31250, 34322, 55778, 57122, 59858, 167042, 171698, 293378, 559682, 916658, 982802, 1062882, 1104098, 1158242, 1195058, 1367858, 1407842, 1414562, 1468898, 1659842, 2380562, 2406818, 2705138, 2789522

Offset: 1

Michel Lagneau, May 30 2016

nonn

a(n) is of the form 2q^2 where q is an odd numbers for which sigma(q^2) is prime (A193070).
The corresponding primes p are 13, 31, 307, 1093, 1723, 2801, 3541, 5113, 8011, 10303, 19531, 17293, 28057, 30941, 30103, 88741, 86143, 147073, 292561, 459007, 492103, 797161, 552793, 579883, 598303, 684757, 704761, 732541, 735307, 830833, 1191373, 1204507, ...
We observe an interesting property: each prime p is element of A053183 (primes of the form m^2 + m + 1 when m is prime) or element of A247837 (primes of the form sigma(2m-1) for a number m) or element of both A053183 and A247837.
Examples:
The numbers 13, 31, 307, 1723, 3541, 5113,... are in A053183;
The numbers 13, 31, 307, 1093, 1723, 2801, 3541,...are in A247837;
The numbers 13, 31, 307, 1723, 3541,... are in A053183 and A247837.

			18 is in the sequence because the divisors of 18 are {1, 2, 3, 6, 9, 18}. The sum of the odd divisors is 1 + 3 + 9 = 13 and the sum of the even divisors is 2 + 6 + 18 = 26 = 2*13.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002384, A053183, A193070, A247837, A278911.

with(numtheory):
for n from 2 by 2  to 500000 do:
   y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
     for k from 1 to n1 do:
       if irem(y[k], 2)=0
        then
        s0:=s0+ y[k]:
        else
        s1:=s1+ y[k]:
      fi:
     od:
     ii:=0:
        if isprime(s1) and s0=2*s1
        then
        printf(`%d, `, n):
         else fi:
     od:

Select[Range[2, 3000000, 2], And[PrimeQ[Total@ Select[#, EvenQ]/2], PrimeQ@ Total@ Select[#, OddQ]] &@ Divisors@ # &] (* Michael De Vlieger, May 30 2016 *)
sodpQ[n_]:=Module[{d=Divisors[n],s},s=Total[Select[d,OddQ]];PrimeQ[ s] && Total[ Select[d,EvenQ]]==2s]; Select[Range[2,279*10^4,2],sodpQ] (* Harvey P. Dale, Dec 01 2020 *)
2 * Select[Range[1, 1200, 2]^2, PrimeQ@DivisorSigma[1, #] &] (* Amiram Eldar, Jul 19 2022 *)

is(n)=my(t); n%4==2 && issquare(n/2,&t) && isprime(n/2+t+1) \\ Charles R Greathouse IV, Jun 08 2016

a(n) >> n^2. - Charles R Greathouse IV, Jun 08 2016

a(n) = 2 * A278911(n) = 2 * A193070(n)^2. - Amiram Eldar, Jul 19 2022

A289356 Least number k such that n^2 + n + k is prime.

Original entry on oeis.org

2, 1, 1, 1, 3, 1, 1, 3, 1, 7, 3, 5, 1, 9, 1, 1, 5, 1, 5, 3, 1, 1, 3, 5, 1, 3, 7, 1, 9, 7, 7, 5, 5, 1, 3, 17, 29, 3, 1, 7, 17, 1, 5, 9, 7, 11, 17, 11, 5, 9, 1, 5, 11, 17, 1, 3, 11, 1, 11, 1, 11, 11, 1, 17, 17, 7, 1, 5, 11, 1, 3, 1, 5, 5, 7, 1, 5, 1, 1, 3, 1, 11, 17, 5, 11, 11

Offset: 0

Gionata Neri, Jul 03 2017

nonn

a(A002384(n)) = 1.

a(A027752(n)) = 3, for n > 2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A002378, A002384, A013632, A027752, A182047 (resulting primes).

seq(nextprime(n^2+n)-(n^2+n), n=0..100); # Robert Israel, Jul 05 2017

Table[k = 1; While[! PrimeQ[n^2 + n + k], k++]; k, {n, 0, 85}] (* Michael De Vlieger, Jul 04 2017 *)

for(n=0,85,k={my(k=1);while(!isprime(n^2+n+k),k++);k;};print1(k", "))

a(n) = A013632(A002378(n)). - Robert Israel, Jul 05 2017

A291689 Numbers n such that n^2 +- n +- 1 are all composite.

Original entry on oeis.org

23, 37, 43, 52, 73, 74, 82, 88, 92, 98, 107, 108, 109, 113, 122, 123, 124, 128, 129, 133, 136, 137, 152, 157, 166, 178, 179, 183, 198, 201, 202, 205, 208, 211, 212, 213, 214, 217, 222, 223, 224, 227, 228, 229, 235, 238, 239, 243, 250, 251, 252, 253, 254, 255, 256, 257, 261, 262, 270, 271, 274

Offset: 1

Robert Israel, Aug 29 2017

nonn

Numbers n such that A291654(n)=1.

Complement of union of A002328, A002384, A045546 and A055494.

			a(1)=23 is in the sequence because 23^2 - 23 - 1 = 505, 23^2 - 23 + 1 = 507, 23^2 + 23 - 1 = 551, 23^2 + 23 + 1 = 553 are all composite.

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

Cf. A002328, A002384, A045546, A055494, A291654.

select(t -> not ormap(isprime, {t^2+t+1,t^2+t-1,t^2-t+1,t^2-t-1}), [$1..1000]);

Select[Range@ 300, Function[t, AllTrue[t^2 + Map[Total[{t, 1} #] &, Tuples[{1, -1}, 2]], ! PrimeQ@ # &]]] (* Michael De Vlieger, Aug 29 2017 *)

is(n)=my(n2=n^2); !isprime(n2+n+1) && !isprime(n2+n-1) && !isprime(n2-n+1) && !isprime(n2-n-1) \\ Charles R Greathouse IV, Aug 30 2017

a(n) ~ n. - Charles R Greathouse IV, Aug 30 2017

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

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

A308316 Numbers m such that q = 2^m - 1 and r = m^2 + m + 1 are both primes.

Original entry on oeis.org

2, 3, 5, 17, 89, 9689, 11213, 2976221

Offset: 1

Jaroslav Krizek, May 19 2019

All terms are primes.
Mersenne exponents p from A000043 such that p^2 + p + 1 is a prime.
Intersection of A000043 and A002384.
Corresponding values of primes q: 3, 7, 31, 131071, 618970019642690137449562111, ...
Corresponding values of primes r: 7, 13, 31, 307, 8011, 93886411, 125742583, 8857894417063, ...

Cf. A000043, A002383, A002384.

[m: m in [1..1000] | IsPrime(2^m - 1)  and IsPrime(m^2 + m + 1)]

isok(n) = isprime(2^n-1) && isprime(n^2+n+1); \\ Michel Marcus, May 21 2019

cp's OEIS Frontend

A089358 Numbers k such that k^2 - 3*k + 3 is prime.

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A117307 Numbers for which (phi(n))^2+phi(n)+1 is a prime number.

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A245476 Least number k > 1 such that k^n + k + 1 is prime, or 0 if no such number exists.

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

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A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.

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A289356 Least number k such that n^2 + n + k is prime.

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Maple

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A291689 Numbers n such that n^2 +- n +- 1 are all composite.

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Maple