A002383 -id:A002383 - cp's OEIS frontend

A239115 Numbers n such that (n-1)*n^2-1 and n^2-(n-1) are both prime.

2, 3, 4, 6, 7, 9, 13, 18, 21, 22, 58, 67, 79, 90, 100, 106, 111, 118, 120, 144, 162, 174, 195, 204, 246, 273, 279, 345, 393, 403, 406, 435, 436, 526, 541, 567, 613, 625, 636, 702, 721, 729, 736, 744, 762, 763, 865, 898, 961, 970, 993, 1059, 1099, 1117, 1131

Offset: 1

Ilya Lopatin following a suggestion from Juri-Stepan Gerasimov, Mar 10 2014,

nonn

Numbers n such that (n^3-n^2-1)*(n^2-n+1) is semiprime.
Intersection of A162293 and A055494.
Primes in this sequence: 2, 3, 7, 13, 67, 79, 541, 613, 1117, ...
Squares in this sequence: 4, 9, 100, 144, 961, ...

			13 is in this sequence because (13-1)*13^2-1 = 2027 and 13^2-(13-1) = 157 are both prime.

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

Cf. A162291, A002383, A239135, A239326.

k:=1;
    for n in [1..1000] do
     if IsPrime(k*(n-1)*n^2-1) and IsPrime(k*n^2-n+1) then
            n;
      end if;
    end for; // Juri-Stepan Gerasimov, Mar 18 2014

Select[Range[1000], PrimeQ[#^3 - #^2 - 1] && PrimeQ[#^2 - # + 1] &] (* Giovanni Resta, Mar 10 2014 *)
Select[Range[1200],PrimeOmega[#^5-2#^4+2#^3-2#^2+#-1]==2&] (* Harvey P. Dale, Sep 24 2014 *)

isok(n) = isprime(n^3-n^2-1) && isprime(n^2-n+1); \\ Michel Marcus, Mar 10 2014

More terms from Giovanni Resta, Mar 10 2014

A093575 Smallest prime of the form n^j+(n+1)^k, with j,k integer > 0, max(j,k)>1.

Original entry on oeis.org

3, 7, 13, 29, 31, 43, 71, 73, 109, 131, 14653, 157, 2393, 211, 241, 83537, 307, 379, 419, 421, 463, 1013, 599, 601, 701, 19709, 757, 615497, 929, 991, 1049537, 2113, 1123, 1259, 2521, 51949, 1481, 1483, 3121, 1721, 1723, 3613, 1979, 2069, 2161, 4977017

Offset: 1

Hugo Pfoertner, Apr 01 2004

nonn

			a(7)=71 because 7+8^2=71 is prime, whereas 7^2+8=57 is composite.

Cf. A002383 primes of form n^2+n+1, A093574, A093576.

A107317 Semiprimes of the form 2*(m^2 + m + 1) (implying that m^2 + m + 1 is a prime).

Original entry on oeis.org

6, 14, 26, 62, 86, 146, 314, 422, 482, 614, 842, 926, 1202, 1514, 2246, 2966, 3446, 5102, 5942, 6614, 7082, 7814, 8846, 9662, 10226, 11402, 12014, 12326, 12962, 16022, 16382, 19802, 20606, 22262, 24422, 24866, 27614, 28562, 34586, 38366, 40046

Offset: 1

Giovanni Teofilatto, May 21 2005

Twice A002383.

Also semiprimes n such that 2*n - 3 is a square. - Giovanni Teofilatto, Dec 29 2005. This coincidence was noticed by Andrew S. Plewe. Proof that this is the same sequence: If X is n^2+(n+1)^2+1, then 2X-3 is 4n^2+4n+1 = (2n+1)^2. And if 2X-3 is a square, then since it's odd, 2X-3 = (2n+1)^2 and X = n^2+(n+1)^2+1. - Don Reble, Apr 18 2007

			a(1)=6 because 1^2 + 2^2 + 1 = 6 = 2*3;
a(2)=14 because 2^2 + 3^2 + 1 = 14 = 2*7;
a(3)=26 because 3^2 + 4^2 + 1 = 26 = 13*2.

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A002383, A002384.

2(#^2 + # + 1) & /@ Select[ Range[144], PrimeQ[ #^2 + # + 1] &] (* Robert G. Wilson v, May 28 2005 *)
fQ[n_] := Plus @@ Last /@ FactorInteger@n == 2 && IntegerQ@Sqrt[2n - 3]; Select[ Range@43513, fQ[ # ] &] (* Robert G. Wilson v *)

for(n=2,100000,if(bigomega(n)==2&&issquare(2*n-3),print1(n,","))) /* Lambert Herrgesell */

a(n) = 2*A002383(n).

a(n) = 2*(A002384(n)^2+A002384(n)+1).

Edited by Robert G. Wilson v, May 28 2005

Re-edited by N. J. A. Sloane, Apr 18 2007

A128878 Primes of the form 47n^2 - 1701n + 10181.

Original entry on oeis.org

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387, 34057, 36821, 39679, 45677, 48817, 52051, 65927, 81307, 89561, 102647, 107197, 116579, 126337, 131357

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Apr 17 2007

nonn

Primes are given in the order in which they arise for increasing n.
Polynomial generates 22 primes for 0 <= n <= 42, i.e., for n = 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42.
If the definition is replaced by "Numbers n of the form 47*k^2 - 1701*k + 10181 such that either n or -n is a prime" we get (essentially) A050267.

			47k^2 - 1701k + 10181 = 21647 for k = 42.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, ISBN 0-387-20860-7, Section A17, page 59.

G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles and Problems Connection.

Cf. A050267, A002383, A027753, A027755, A005471, A027758, A048059, A007635, A005846, A116206, A050268, A022464.

Select[Table[47*n^2 - 1701*n + 10181, {n, 0, 100}], # > 0 && PrimeQ[#] &] (* T. D. Noe, Aug 02 2011 *)

Edited by Klaus Brockhaus, Apr 22 2007 and by N. J. A. Sloane, May 05 2007 and May 06 2007

A174967 Smallest number of the form k^2 + k + 1 with n distinct prime divisors.

Original entry on oeis.org

1, 3, 21, 273, 10101, 316407, 6914271, 2424626841, 346084535811, 6177672967557, 1741866776384007, 92264158181274807, 103008522046409631057, 22810816825458528984663, 2220066397007943013450011, 545889722100356705628041121, 73293936170018923619553695493

Offset: 0

Michel Lagneau, Apr 02 2010

nonn

If k == 2 (mod 3), all prime divisors of k^2 + k + 1 are congruent to 1 (mod 3), and if k == 1 (mod 3), the number 3 is divisor, and the other divisors are congruent to 1 (mod 3).
Proof: first case: k == 2 (mod 3): let q divide k^2 + k + 1. Then 4q divides 4*(k^2 + k + 1) = (2k+1)^2 + 3, and (-3/q)=1, where (a/b) is the Legendre symbol. By using the law of quadratic reciprocity, we obtain (-3/q) = (-1/q)(3/q) = (-1/q)(q/3)(-1)^((q-1)/2)(3-1)/2)) = ((-1)^(q-1)/2)((-1)^(q-1)/2)(q/3) = (q/3). Suppose q !== 1 (mod 3). Then k^2 + k + 1 !== 0 (mod 3) => q == 2 (mod 3), and then (q/3) = -1 => (-3/q) = -1, a contradiction. So q == 1 (mod 3).
Second case: k == 1 (mod 3) => 3 is divisor of k^2 + k + 1, and the other divisors q == 1 (mod 3).
a(11) <= 4943071434145592163, a(12) <= 2702887058650660754061, a(13) <= 896265629366361887178273, a(14) <= 72053193593257327979705541. - Michael S. Branicky, Mar 21 2021
Is a(n) squarefree? The first 16 terms are. - David A. Corneth, Mar 21 2021

			21 = 3*7;
273 = 3*7*13;
10101 = 3*7*13*37;
316407 = 3*7*13*19*61;
6914271 = 3*7*13*19*31*43;
2424626841 = 3*7*13*19*61*79*97;
346084535811 = 3*7*19*37*43*67*79*103;
6177672967557 = 3*7*13*19*31*43*61*97*151;
1741866776384007 = 3*7*13*19*31*37*43*67*151*673.

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.

Cf. A002384, A002383.

A174967 := proc(n)
        for k from 1 do
                a := k^2+k+1 ;
                if A001221(a) = n then
                        return a;
                end if;
        end do:
end proc: # R. J. Mathar, Jul 06 2012

from sympy import primefactors
def a(n):
  k = 1
  while len(primefactors(k**2 + k + 1)) != n: k += 1
  return k**2 + k + 1
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Mar 21 2021

a(10) from Michael S. Branicky, Mar 21 2021

a(0), a(11)-a(16) from David A. Corneth, Mar 21 2021

A249622 a(n) = number of ways to express A117048(n) as the sum of two positive triangular numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 6, 2, 1, 1, 1, 1

Offset: 1

Zak Seidov, Nov 03 2014

nonn

			a(6) = 2 because A117048(6) = 31 and 31 = 3 + 28 = 10 + 21 (first case of two-way expression).
a(22) = 3 because A117048(22) = 181 and 181 = A000217(i) + A000217(k), for {i,k} = {{4, 18}, {7, 17}, {9, 16}} (first case of three-way expression): 181 = 10 + 171 = 28 + 153 = 45 + 136.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217, A002383.

A265006 Twin prime pairs of the form (k^2 + k - 1, k^2 + k + 1).

Original entry on oeis.org

5, 7, 11, 13, 29, 31, 41, 43, 71, 73, 239, 241, 419, 421, 461, 463, 599, 601, 1481, 1483, 1721, 1723, 2549, 2551, 2969, 2971, 3539, 3541, 4421, 4423, 8009, 8011, 10301, 10303, 17291, 17293, 19181, 19183, 20021, 20023, 23561, 23563, 24179, 24181, 27059, 27061, 31151, 31153, 35531, 35533

Offset: 1

Bill McEachen, Nov 29 2015

This is a subset of A002327 and A002383 taken together. Note that 3 is not a member, as the pairing (3, 5) is excluded as defined, as 3 and 5 associate to different centers.
The corresponding n are in A088485.
The average of each twin prime pair is an oblong number (A002378). - Michel Marcus, Feb 04 2017

			For k = 6, k^2 + k = 6^2 + 6 = 42, and (41,43) is a twin prime pair, so 41 and 43 are in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3594 from G. C. Greubel)

Cf. A002327, A002378, A002383, A088485, A088486, A306889.

&cat[[n^2+n-1, n^2+n+1]: n in [0..250]| IsPrime(n^2+n-1) and IsPrime(n^2+n+1)]; // Vincenzo Librandi, Feb 05 2017

{#^2 + # - 1, #^2 + # + 1} & /@ Select[Range@ 200, PrimeQ[#^2 + # - 1] && PrimeQ[#^2 + # + 1] &] // Flatten (* Michael De Vlieger, Nov 30 2015 *)
Flatten[Select[Table[n^2 + n + {-1, 1}, {n, 0, 200}], And@@PrimeQ[#] &]] (* Vincenzo Librandi, Feb 05 2017 *)

PARI
```
genit()={my(maxx=1000);n=0;while(n
				
```

a(2n-1) = A088486(n). a(2n)=2+a(2n-1).

A272571 Primes of the form n^4 + n + 1 with n positive.

Original entry on oeis.org

3, 19, 631, 1303, 6571, 14653, 20749, 38431, 331801, 457003, 1048609, 1679653, 3748141, 4879729, 12960061, 22667191, 26873929, 29986651, 35153119, 62742331, 65610091, 108243319, 131079709, 200534041, 252047503, 294500053, 454372003, 466949029, 639129121

Offset: 1

Vincenzo Librandi, May 03 2016

nonn

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

Cf. A002383, A245488, A272572.

[a: n in [0..200] | IsPrime(a) where a is n^4+n+1];

Select[Table[n^4 + n + 1, {n, 100}], PrimeQ]

lista(nn) = {for(n=1, nn, if(ispseudoprime(p=n^4+n+1), print1(p, ", "))); } \\ Altug Alkan, May 04 2016

A272572 Primes of the form n^4 + n^3 + 1 with n positive.

Original entry on oeis.org

3, 109, 751, 15973, 41161, 54001, 345601, 1543501, 1726273, 2372761, 3833281, 8039359, 10010113, 18125251, 25769593, 63447211, 75609559, 93178009, 147741001, 164490259, 170377561, 270532609, 432967681, 457483993, 509625001, 551562859, 596037313, 1055592001

Offset: 1

Vincenzo Librandi, May 03 2016

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

Cf. A002383, A245488, A272571.

[a: n in [0..200] | IsPrime(a) where a is n^4+n^3+1];

Select[Table[n^4 + n^3 + 1, {n, 0, 170}], PrimeQ]

lista(nn) = {for(n=1, nn, if(ispseudoprime(p=n^4+n^3+1), print1(p, ", "))); } \\ Altug Alkan, May 04 2016

A343774 Primes of the form (c^k+1)/(c+1) not having a representation in the form (b^q-1)/(b-1), where b, c > 1 and k, q > 2.

Original entry on oeis.org

3, 11, 61, 521, 547, 683, 2731, 9091, 13421, 19141, 43691, 61681, 152381, 174763, 185641, 224071, 398581, 909091, 1151041, 1623931, 1824841, 2031671, 2796203, 3341101, 4778021, 5200081, 7027567, 8987221, 10678711, 15790321, 22796593, 25058741, 31224301, 32222107

Offset: 1

Bernard Schott, Apr 29 2021

The exponents k, q are necessarily primes.
Equivalently: primes of the form (c^k+1)/(c+1) that are not Brazilian: intersection of A059055 and A220627.
Except for 3 where k = 3, all the terms of this sequence are of the form (c^k+1)/(c+1) with k prime >= 5.
The only known prime of this form with k prime >= 5 that is not present is 43 = (2^7+1)/(2+1) because also 43 = (7^3+1)/(7+1) = (6^3-1)/(6-1) = 111_6, so 43 belongs to A002383.

			3 = (2^3+1)/(2+1) is not Brazilian, hence 3 is a term.
11 = (2^5+1)/(2+1) is not Brazilian, hence 11 is a term.
547 = (3^7+1)/(3+1) is not Brazilian, hence 547 is a term.
9091 = (10^5+1)/(10+1) is not Brazilian, hence 9091 is a term.

H. Dubner and T. Granlund, Primes of the form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.

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 (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) = this sequence.
Primes neither of the form (c^q+1)/(c+1) nor (b^k-1)/(b-1) = A343775.

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

More terms from Michel Marcus, Apr 30 2021

cp's OEIS Frontend

A239115 Numbers n such that (n-1)*n^2-1 and n^2-(n-1) are both prime.

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A093575 Smallest prime of the form n^j+(n+1)^k, with j,k integer > 0, max(j,k)>1.

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A107317 Semiprimes of the form 2*(m^2 + m + 1) (implying that m^2 + m + 1 is a prime).

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A128878 Primes of the form 47*n^2 - 1701*n + 10181.

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A174967 Smallest number of the form k^2 + k + 1 with n distinct prime divisors.

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A249622 a(n) = number of ways to express A117048(n) as the sum of two positive triangular numbers.

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A265006 Twin prime pairs of the form (k^2 + k - 1, k^2 + k + 1).

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Magma

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A272571 Primes of the form n^4 + n + 1 with n positive.

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A128878 Primes of the form 47n^2 - 1701n + 10181.