A028870 -id:A028870 - cp's OEIS frontend

A293620 Numbers k such that f(k), f(k+1) and f(k+2) are all primes, where f(k) = (2k+1)^2 - 2 (A073577).

1, 2, 16, 58, 149, 177, 534, 681, 954, 1045, 1052, 1255, 1367, 1563, 2046, 2074, 2515, 2557, 2564, 2788, 3586, 3593, 3908, 4062, 4552, 5252, 5371, 5385, 6400, 6729, 7443, 7478, 9305, 9375, 9942, 10355, 10411, 10726, 10740, 11286, 11545, 11559, 11832, 11965

Offset: 1

Amiram Eldar, Oct 13 2017

nonn

Sierpiński proved that under Schinzel's hypothesis H this sequence is infinite.
Sierpiński showed that the only quadruple of consecutive primes of the form (2k+1)^2 - 2 are for k = 1 (i.e., 1 and 2 are the only consecutive terms in this sequence).
Numbers k such that the 3 consecutive integers k, k+1 and k+2 belong to A088572. - Michel Marcus, Oct 13 2017

			The first triples are: k = 1: (7, 23, 47), k = 2: (23, 47, 79), k = 16: (1087, 1223, 1367).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wacław Sierpiński, Remarque sur la distribution de nombres premiers, Matematički Vesnik, Vol. 2(17), Issue 31 (1965), pp. 77-78.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Wikipedia, Schinzel's hypothesis H.

Cf. A088572, A008865, A028870, A028871, A073577, A088572.

Select[Range[10^4], AllTrue[{(2#+1)^2-2, (2#+3)^2-2, (2#+5)^2-2},PrimeQ] &]
SequencePosition[Table[If[PrimeQ[(2k+1)^2-2],1,0],{k,12000}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Feb 09 2025 *)

f(n) = 4*n^2 + 4*n - 1;
isok(n) = isprime(f(n)) && isprime(f(n+1)) && isprime(f(n+2)); \\ Michel Marcus, Oct 13 2017

A309726 Numbers k such that k^2 - 12 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 25, 29, 35, 41, 49, 53, 59, 61, 79, 85, 91, 95, 97, 103, 107, 113, 119, 121, 137, 139, 145, 149, 163, 169, 173, 179, 181, 185, 191, 205, 209, 227, 233, 235, 245, 251

Offset: 1

Daniel Starodubtsev, Aug 14 2019

nonn

All terms are odd and not divisible by 3.

			11 is in the sequence because 11^2 - 12 = 109, which is prime.

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

Cf. A028870, A028873, A028876, A028879, A028882, A028885, A186815, A090696, A296507.

Cf. A056927.

Select[Range[5,301,2],PrimeQ[#^2-12]&] (* Harvey P. Dale, Dec 23 2019 *)

select(n->isprime(n^2-12), [1..1000]) \\ Andrew Howroyd, Aug 14 2019

If A056927(k) = 12, then k is a term. - A.H.M. Smeets, Aug 15 2019

A328525 Numbers k such that (k-1)k(k+1) = (k-1)(1+u) = k(1+v) = (k+1)*(1+w) with primes u, v, w.

Original entry on oeis.org

3, 5, 9, 21, 55, 131, 145, 155, 231, 259, 265, 449, 495, 561, 595, 1045, 1051, 1365, 1409, 1491, 1549, 1849, 1989, 2001, 2101, 2469, 2785, 3365, 3621, 3641, 3669, 3845, 3911, 4285, 4951, 5181, 5465, 6049, 6699, 7189, 7229, 8219, 8629, 9175, 9521, 9539, 9631, 9729

Offset: 1

Frank Ellermann, Feb 24 2020

nonn

			3 is a term because 2*3*4 = 2*(1+11) = 3*(1+7) = 4*(1+5) with primes 11, 7, 5.
9 is a term because 8*9*10 = 8*(1+89) = 9*(1+79) = 10*(1+71) with primes 89, 79, 71.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040.

Intersection of A002328, A028870 and A045546.

q:= k-> andmap(isprime, (t-> [t-1, t-k, t+k])(k^2-1)):
select(q, [$1..10000])[];  # Alois P. Heinz, Feb 25 2020

Select[Range[2, 10^4], AllTrue[{(# - 1)*#, #*(# + 1), (# + 1)*(# - 1)} - 1, PrimeQ] &] (* Amiram Eldar, Feb 24 2020 *)

isok(k) = isprime(k*(k+1)-1) && isprime((k+1)*(k-1)-1) && isprime(k*(k-1)-1); \\ Michel Marcus, Feb 25 2020

S = 3
do N = 5 to 595 by 2
   if NOPRIME( N*N +N -1 ) then  iterate N
   if NOPRIME( N*N    -2 ) then  iterate N
   if NOPRIME( N*N -N -1 ) then  iterate N
   S = S || ',' N
end N
say S

More terms from Amiram Eldar, Feb 24 2020

A330438 Numbers k such that k^2-2 and k^3-2 are prime.

Original entry on oeis.org

9, 15, 19, 27, 37, 121, 135, 145, 211, 217, 259, 265, 267, 279, 355, 357, 387, 391, 435, 489, 525, 561, 615, 621, 727, 951, 987, 1029, 1119, 1141, 1177, 1251, 1287, 1357, 1435, 1491, 1561, 1617, 1717, 1785, 1819, 1839, 1875, 1909, 1989, 2001, 2077, 2107, 2211

Offset: 1

K. D. Bajpai, Dec 14 2019

nonn

Intersection of A028870 and A038599.

			a(1) = 9: 9^2 - 2 = 79; 9^3 - 2 = 727; both results are prime.
a(2) = 15: 15^2 - 2 = 223; 15^3 - 2 = 3373; both results are prime.

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

Cf. A028870, A028873, A038599, A038600, A108701.

[n : n in [1 .. 100] | IsPrime (n^2 - 2) and IsPrime (n^3 - 2)];

filter:= k -> isprime(k^2-2) and isprime(k^3-2):
select(filter, [$2..10000]); # Robert Israel, Dec 24 2019

Select[Range[10000], PrimeQ[#^3 - 2] && PrimeQ[#^2 - 2] &]

A239475 Least number k such that k^n + n and k^n - n are both prime, or 0 if no such number exists.

Original entry on oeis.org

4, 3, 2, 0, 42, 175, 66, 3, 2, 4983, 1770, 55055, 28686, 18765, 8456, 0, 594, 128345, 136080, 81, 92, 1163409, 18810, 10415, 11754, 3855, 0, 86043, 38880, 17639, 26088, 37293, 5540, 612015, 6876, 0, 44220, 130425, 110, 9292527, 1004850, 1812149, 442404, 1007445, 570658

Offset: 1

Derek Orr, Mar 20 2014

nonn

a(n) = 0 iff n is of the form (pk)^p for some k and some prime p (See A097764).

gcd(n,a(n)) = 1 for all a(n) > 0.

			1^1 +/- 1 = 2 and 0 are not both primes. 2^1 +/- 1 = 3 and 1 are not both primes. 3^1 +/- 1 = 4 and 2 are not both primes. 4^1 +/- 1 = 5 and 3 are both primes. Thus a(1) = 4.

Cf. A072883, A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239418, A239474.

a(n)=for(k=1,10^7,if(ispseudoprime(k^n-n)&&ispseudoprime(k^n+n),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

import sympy
from sympy import isprime
def TwoBoth(x):
  for k in range(1,10**7):
    if isprime(k**x+x) and isprime(k**x-x):
      return k
x = 1
while x < 100:
  if TwoBoth(x) != None:
    print(TwoBoth(x))
  else:
    print(0)
  x += 1

a(A097764(n)) = 0 for all n.

A334294 Numbers k such that 70k^2 + 70k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 45, 46, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 74, 76, 77, 78, 79, 80, 81, 82, 87, 88, 90, 93, 96, 97, 100

Offset: 1

James R. Buddenhagen, Apr 21 2020

nonn

Among quadratic polynomials in k of the form a*k^2 + a*k - 1 the value a=70 gives the most primes for any a in the range 1<=a<=300, at least up to k=40000. Here a and k are positive integers. Other "good" values of a are a=250, a=99, and a=19.

			For k=1, 70*k^2 + 70*k - 1 = 70*1^2 + 70*1 - 1 = 139, which is prime, so 1 is in the sequence.

N. Boston et M. L.Greenwood, Quadratics representing primes, Amer. Math. Monthly 102:7 (1995), 595-599.
François Dress and Michel Olivier, Polynômes prenant des valeurs premières, Experimental Mathematics, Volume 8, Issue 4 (1999), 319-338.
G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Cf. A001912, A002384, A005574, A027861, A028870, A056900.

Cf. A067201, A088572, A089623, A091199, A271980.

a:=proc(n) if isprime(70*n^2+70*n-1) then n else NULL end if end proc;
seq(a(n),n=1..100);

Select[Range@ 100, PrimeQ[70 #^2 + 70 # - 1] &] (* Michael De Vlieger, May 26 2020 *)

cp's OEIS Frontend

A293620 Numbers k such that f(k), f(k+1) and f(k+2) are all primes, where f(k) = (2k+1)^2 - 2 (A073577).

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A309726 Numbers k such that k^2 - 12 is prime.

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A328525 Numbers k such that (k-1)*k*(k+1) = (k-1)*(1+u) = k*(1+v) = (k+1)*(1+w) with primes u, v, w.

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A330438 Numbers k such that k^2-2 and k^3-2 are prime.

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A239475 Least number k such that k^n + n and k^n - n are both prime, or 0 if no such number exists.

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A334294 Numbers k such that 70*k^2 + 70*k - 1 is prime.

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Maple

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A328525 Numbers k such that (k-1)k(k+1) = (k-1)(1+u) = k(1+v) = (k+1)*(1+w) with primes u, v, w.

A334294 Numbers k such that 70k^2 + 70k - 1 is prime.