A005098 Numbers k such that 4k + 1 is prime.

1, 3, 4, 7, 9, 10, 13, 15, 18, 22, 24, 25, 27, 28, 34, 37, 39, 43, 45, 48, 49, 57, 58, 60, 64, 67, 69, 70, 73, 78, 79, 84, 87, 88, 93, 97, 99, 100, 102, 105, 108, 112, 114, 115, 127, 130, 135, 139, 142, 144, 148, 150, 153, 154, 160, 163, 165, 168, 169, 175, 177, 183

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Sum of i-th and j-th triangular numbers, where i=A096029(n), j=A096030(n); i.e., a(n) = A000217(A096029(n)) + A000217(A096030(n)). - Lekraj Beedassy, Jun 16 2004

For every k in the sequence, there is exactly 1 square number that can be subtracted to leave a pronic (A002378). E.g., 27 - 25 = 2, 99 - 9 = 90. - Jon Perry, Nov 06 2010

See A208295 for details concerning the preceding Jon Perry comment. - Wolfdieter Lang, Mar 29 2012

a(k) appears in the o.g.f. for floor(A002144(k)*j^2/4), j >= 0, for k >= 1: x*(a(k)*(1 + x^2) + b(k)*x)/((1 - x)^3*(1 + x)), together with b(k) = (A002144(k) + 1)/2 = A119681(k). - Wolfdieter Lang, Aug 07 2013

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Eric Weisstein's World of Mathematics, Wilson's Theorem.

Crossrefs

See A002144 for the actual primes.

Cf. A002972, A002973, A173331, A173330, A023212.

Programs

Haskell

a005098 = (`div` 4) . (subtract 1) . a002144
-- Reinhard Zumkeller, Mar 17 2013

Magma

[k: k in [0..10000] | IsPrime(4*k+1)] // Vincenzo Librandi, Nov 18 2010

Maple

a := []; for k from 1 to 500 do if isprime(4*k+1) then a := [op(a), k]; fi; od: A005098 := k->a[k];

Mathematica

Select[Range[200], PrimeQ[4# + 1] &] (* Harvey P. Dale, Apr 20 2011 *)

PARI

is(k)=isprime(4*k+1) \\ Charles R Greathouse IV, Nov 20 2012

Formula

a(n) = (A002144(n)-1)/4.

Extensions

More terms from Ray Chandler, Jun 26 2004

Edited by Charles R Greathouse IV, Mar 17 2010