A285790 -id:A285790 - cp's OEIS frontend

A176616 Primes of the form x^2 + 7*y^2, where x and y=x+1 are consecutive natural numbers.

7, 29, 67, 191, 277, 379, 631, 947, 1129, 1327, 2017, 2557, 2851, 4561, 4951, 5779, 6217, 8647, 9181, 12721, 13367, 14029, 15401, 16111, 17579, 20707, 21529, 22367, 24091, 24977, 31627, 36857, 37951, 42487, 43661, 44851, 47279, 53629, 58997

Offset: 1

Giovanni Teofilatto, Apr 22 2010

nonn

Primes of the form 8*n^2+14*n+7 = (2*n+2)*(4*n+3)+1 = A000384(2*n+2)+1. - Vincenzo Librandi, Apr 25 2010

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

Cf. A176608.

A285790 is an almost identical sequence.

[ a: n in [0..250] | IsPrime(a) where a is 8*n^2+14*n+7 ] // Vincenzo Librandi, Apr 25 2010

Select[Table[x^2+7(x+1)^2,{x,0,100}],PrimeQ] (* Harvey P. Dale, May 03 2024 *)

Definition made more accurate by R. J. Mathar, May 04 2010
Corrected (inserted 7) and extended by Vincenzo Librandi, Apr 25 2010
Offset corrected by Mohammed Yaseen, May 20 2023

A285792 Primes equal to an octagonal number plus 1.

Original entry on oeis.org

2, 41, 97, 281, 409, 937, 1409, 2297, 4721, 5209, 6257, 8009, 8641, 12161, 14561, 18097, 21001, 23057, 24121, 26321, 27457, 37409, 42961, 50441, 52009, 55217, 56857, 60209, 70841, 76481, 90481, 139537, 147409, 152777, 161009, 169457, 172321, 227977, 238009

Offset: 1

Colin Barker, Apr 26 2017

nonn

Primes in A056105. - Omar E. Pol, Apr 26 2017

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000040, A000567, A002496, A055469, A285789, A285790, A285791.

Select[PolygonalNumber[8,Range[300]]+1,PrimeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 22 2017 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
maxk=300; L=List(); for(k=1, maxk, if(isprime(p=pg(8, k) + 1), listput(L, p))); Vec(L)

A285789 Primes equal to a pentagonal number plus 1.

Original entry on oeis.org

2, 13, 23, 71, 211, 331, 853, 1163, 1427, 2381, 2753, 3433, 3877, 5923, 6113, 6701, 7741, 8627, 9323, 11311, 12377, 14653, 14951, 17443, 18427, 23003, 27271, 31033, 32341, 32783, 34127, 38321, 43777, 52361, 55201, 56941, 57527, 62323, 64171, 64793, 69877

Offset: 1

Colin Barker, Apr 26 2017

nonn

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000040, A000326, A002496, A055469, A285790, A285791, A285792.

Select[PolygonalNumber[5,Range[300]]+1,PrimeQ] (* Harvey P. Dale, Dec 04 2024 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
maxk=300; L=List(); for(k=1, maxk, if(isprime(p=pg(5, k) + 1), listput(L, p))); Vec(L)

A285791 Primes equal to a heptagonal number plus 1.

Original entry on oeis.org

2, 19, 113, 149, 541, 617, 971, 1289, 1783, 2357, 3011, 3187, 5689, 6427, 7481, 7757, 9829, 12497, 12853, 14327, 15881, 17099, 18793, 21023, 24851, 28463, 30637, 31193, 45361, 50909, 54539, 60607, 63761, 66179, 69473, 70309, 83449, 88079, 90917, 91873, 94771

Offset: 1

Colin Barker, Apr 26 2017

nonn

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000040, A000566, A002496, A055469, A285789, A285790, A285792.

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
maxk=300; L=List(); for(k=1, maxk, if(isprime(p=pg(7, k) + 1), listput(L, p))); Vec(L)

from sympy import isprime
def heptagonal(n): return n*(5*n-3)//2
def aupto(limit):
  alst, n, hn = [], 1, heptagonal(1)
  while hn < limit:
    if isprime(hn+1): alst.append(hn+1)
    n, hn = n+1, heptagonal(n+1)
  return alst
print(aupto(94771)) # Michael S. Branicky, Feb 19 2021

cp's OEIS Frontend

A176616 Primes of the form x^2 + 7*y^2, where x and y=x+1 are consecutive natural numbers.

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A285792 Primes equal to an octagonal number plus 1.

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A285789 Primes equal to a pentagonal number plus 1.

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Mathematica

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A285791 Primes equal to a heptagonal number plus 1.

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Python