A088572 -id:A088572 - cp's OEIS frontend

A089623 Numbers n such that n^2 + 2n - 1 is prime.

1, 2, 4, 6, 8, 12, 14, 18, 20, 26, 28, 32, 34, 36, 42, 46, 48, 54, 60, 62, 68, 70, 74, 76, 88, 92, 102, 106, 116, 118, 120, 126, 130, 134, 138, 144, 154, 160, 168, 172, 176, 182, 190, 204, 210, 216, 222, 230, 232, 236, 238, 246, 252, 256, 258, 264, 266, 272

Offset: 1

Giovanni Teofilatto, Dec 31 2003

Equivalently, numbers n such that (n+1)^2 - 2 is prime.

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.

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

Cf. A088572. See A028871 for the actual primes.

[n: n in [0..400] | IsPrime(n^2+2*n-1)]; // Vincenzo Librandi, Dec 17 2010

Select[Range[1000], PrimeQ[#^2 + 2 # - 1] &] (* Vincenzo Librandi, May 24 2014 *)

is(n)=isprime(n^2+2*n-1) \\ Charles R Greathouse IV, Jun 05 2017

a(n) = 2*A088572(n).

Corrected (1 prepended and 96 replaced with 92) by Vincenzo Librandi, Dec 17 2010

A225844 Least k>0 such that triangular(n) + k*(k+1) is a triangular number.

Original entry on oeis.org

2, 1, 3, 5, 7, 2, 11, 13, 5, 17, 19, 3, 6, 25, 27, 9, 31, 33, 35, 4, 9, 41, 8, 45, 47, 10, 14, 53, 9, 5, 59, 61, 21, 18, 67, 69, 21, 73, 75, 14, 22, 6, 11, 13, 87, 15, 91, 26, 20, 34, 12, 101, 26, 105, 30, 7, 20, 33, 115, 117, 119, 34, 21, 125, 37, 129, 29, 133, 14, 137

Offset: 0

Alex Ratushnyak, May 17 2013

nonn

For n>0, a(n) <= 2*n-1, because n*(n+1)/2 + (2*n-1)*2*n =  (9*n^2 - 3*n)/2 = 3*n*(3*n-1)/2 = triangular(3*n-1).
The subsequence with terms less than 2*n-1 begins: 2, 5, 3, 6, 9, 4, 9, 8, 10, 14, 9, 5, 21, 18, 21, 14, 22, 6, 11, 13, 15, ...
The sequence of n's such that a(n) < 2*n-1 begins: 5, 8, 11, 12, 15, 19, 20, 22, 25, 26, ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000217, A002378, A082183, A088572.

Cf. A101157 (least k>0 such that triangular(n) + k^2 is a triangular number).

a:= proc(n) option remember; local w, k; w:= n*(n+1)/2;
      for k while not issqr(8*(w+k*(k+1))+1) do od; k
    end:
seq(a(n), n=0..69);  # Alois P. Heinz, Nov 13 2024

lktrno[n_]:=Module[{t=(n(n+1))/2,k=1},While[!IntegerQ[(Sqrt[ 8(t+k(k+1))+1]-1)/2],k++];k]; Array[lktrno,70,0] (* Harvey P. Dale, Aug 19 2014 *)

a(n)=for(k=1,2*n,t=n*(n+1)/2+k*(k+1);x=sqrtint(2*t);if(t==x*(x+1)/2,return(k))) /* from Ralf Stephan */

def isTriangular(a):
    sr = 1 << (a.bit_length() >> 1)
    a += a
    while a < sr*(sr+1):  sr>>=1
    b = sr>>1
    while b:
      s = sr+b
      if a >= s*(s+1):  sr = s
      b>>=1
    return (a==sr*(sr+1))
n = tn = 0
while 1:
  for m in range(1, 1000000000):
    if isTriangular(tn + m*(m+1)): break
  print(m, end=', ')
  n += 1
  tn += n

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).

Original entry on oeis.org

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

A323741 a(n) = m-p where m = (2n+1)^2 and p is the largest prime < m.

Original entry on oeis.org

2, 2, 2, 2, 8, 2, 2, 6, 2, 2, 6, 6, 2, 2, 8, 2, 2, 2, 10, 12, 2, 8, 2, 2, 8, 6, 2, 20, 12, 2, 2, 6, 6, 2, 2, 6, 2, 2, 12, 8, 6, 6, 8, 2, 8, 2, 12, 6, 10, 8, 2, 22, 2, 14, 20, 6, 6, 2, 2, 2, 8, 6, 2, 8, 2, 6, 2, 12, 2, 14, 6, 2, 8, 8, 14, 10, 2, 18, 20, 2, 8, 14, 6, 2, 10, 2, 32, 2, 12, 12, 2, 8, 6, 44, 2, 6, 14, 6, 20, 14

Offset: 1

Ali Sada, Sep 03 2019

nonn

a(n) cannot be a square: suppose a(n) = k^2; then p=m-a(n) could be factored as (2n+k-1)*(2n-k-1); hence it would not be a prime.
Legendre's conjecture implies a(n) <= 4*n.  Oppermann's conjecture implies a(n) <= 2*n. - Robert Israel, Sep 04 2019
All terms are even. - Alois P. Heinz, Sep 04 2019

			When n=4, m=81, p=79, so a(4) = 81-79 = 2.

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

Cf. A016754, A049711, A088572, A089166, A151799.

seq((2*n+1)^2-prevprime((2*n+1)^2),n=1..100); # Robert Israel, Sep 04 2019

mp[n_]:=Module[{m=(2n+1)^2},m-NextPrime[m,-1]]; Array[mp, 100] (* Harvey P. Dale, Feb 03 2022 *)

a(n) = (2*n+1)^2 - precprime((2*n+1)^2 - 1); \\ Michel Marcus, Sep 05 2019

a(n) = A049711(A016754(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 *)

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A089623 Numbers n such that n^2 + 2n - 1 is prime.

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A225844 Least k>0 such that triangular(n) + k*(k+1) is a triangular number.

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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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A323741 a(n) = m-p where m = (2n+1)^2 and p is the largest prime < m.

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

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