A084695 Triangle read by rows in which row n lists the n smallest positive numbers k such that k + n is a prime.

1, 1, 3, 2, 4, 8, 1, 3, 7, 9, 2, 6, 8, 12, 14, 1, 5, 7, 11, 13, 17, 4, 6, 10, 12, 16, 22, 24, 3, 5, 9, 11, 15, 21, 23, 29, 2, 4, 8, 10, 14, 20, 22, 28, 32, 1, 3, 7, 9, 13, 19, 21, 27, 31, 33, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 1, 5, 7, 11, 17, 19, 25, 29, 31, 35, 41, 47

Offset: 1

Author

Amarnath Murthy and Jason Earls, Jul 12 2003

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,  3;
  2,  4,  8;
  1,  3,  7,  9;
  2,  6,  8, 12, 14;
  1,  5,  7, 11, 13, 17;
  4,  6, 10, 12, 16, 22, 24;
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

First column gives A013632, last gives A084747.

Cf. A000040, A000720.

Programs

Magma

[NthPrime(#PrimesUpTo(n) +k) -n: k in [1..n], n in [1..16]]; // G. C. Greubel, May 12 2023

Mathematica

nn=30;Flatten[With[{prs=Prime[Range[nn]]},Table[Take[prs,{PrimePi[n]+1, PrimePi[n]+n}]-n,{n,Floor[nn/2]}]]] (* Harvey P. Dale, Dec 07 2012 *)
Table[Prime[PrimePi[n] +k] -n, {n,16}, {k,n}]//Flatten (* G. C. Greubel, May 12 2023 *)

SageMath

def A084695(n,k): return nth_prime(prime_pi(n) + k) - n
flatten([[A084695(n,k) for k in range(1,n+1)] for n in range(1,17)]) # G. C. Greubel, May 12 2023

Formula

T(n, k) = prime(PrimePi(n) + k) - n. - G. C. Greubel, May 12 2023