A187677 Primes of the form 8*k^2 + 6*k - 1 for positive k.

13, 43, 89, 151, 229, 433, 701, 859, 1033, 1223, 1429, 1889, 2143, 2699, 3001, 3319, 4003, 4751, 5563, 7873, 10009, 11173, 11779, 12401, 13693, 17203, 18719, 19501, 21943, 25423, 27259, 28201, 30133, 31123, 33151, 36313, 38501, 39619, 41903, 46663, 49139, 51679

Offset

1,1

Comments

In a variant of the Ulam spiral in which only odd numbers are entered, some primes still line up along some diagonals but not others. Without the even numbers, primes can also line up in horizontal and diagonal lines. This sequence comes from an upwards vertical line which starts with 13.

Primes of A091823. - Klaus Purath, Jan 03 2021

This is a subsequence of A162761. - Davide Rotondo, Jun 14 2025

Links

Table of n, a(n) for n=1..42.

Alonso del Arte, Ulam spiral (2009). [EmdrGreg's comment suggested the odd number spiral variant.]

OEIS Wiki, Ulam spiral

Formula

a(n) = 2((2n - 1)^2 - n) - 1 (or, find the number in the corresponding spot in the better-known Ulam spiral, double it and subtract 1).

The polynomial 8n^2 - 10n + 1 produces the same primes.

Mathematica

Select[Table[2((2n - 1)^2 - n) - 1, {n, 100}], PrimeQ]

Prog

(Magma) [ a: n in [0..2500] | IsPrime(a) where a is 8*n^2 + 6*n - 1 ]; // Vincenzo Librandi, Apr 24 2011
(PARI) lista(nn) = my(list=List(), p); for (n=1, nn, if(isprime(p=8*n^2+6*n-1), listput(list, p))); Vec(list); \\ Michel Marcus, Jun 14 2025

Crossrefs

Cf. A073337 and A168026 are diagonals of the usual Ulam spiral which have some of the same primes as this vertical line.

Cf. A091823, A162761.

Sequence in context: A282322 A031382 A385824 * A082040 A106734 A066465

Adjacent sequences: A187674 A187675 A187676 * A187678 A187679 A187680

Keyword

easy,nonn

Author

Alonso del Arte, Mar 21 2011

Status

approved