A153270 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 2, read by rows.

3, 3, 12, 3, 15, 105, 3, 18, 162, 1944, 3, 21, 231, 3465, 65835, 3, 24, 312, 5616, 129168, 3616704, 3, 27, 405, 8505, 229635, 7577955, 295540245, 3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800, 3, 33, 627, 16929, 592515, 25478145, 1299385395, 76663738305, 5136470466435

Offset: 0

Author

Roger L. Bagula, Dec 22 2008

Keywords

• nonn
• tabl

Comments

Row sums are {3, 15, 123, 2127, 69555, 3751827, 303356775, 34403458143, 5214459678387, 1018396843935195, 249088654250968899, ...}.

Examples

			Triangle begins as:
  3;
  3, 12;
  3, 15, 105;
  3, 18, 162,  1944;
  3, 21, 231,  3465,  65835;
  3, 24, 312,  5616, 129168,  3616704;
  3, 27, 405,  8505, 229635,  7577955, 295540245;
  3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. this sequence (m=2), A153271 (m=3), A153272 (m=4).

Cf. A000142, A001710, A001813, A008544, A008545.

Cf. A001147, A032031, A047056, A144739, A144758.

Programs

Magma

m:=2;
function T(n,k)
  if k eq 0 then return NthPrime(m);
  else return (&*[j*n + NthPrime(m): j in [0..k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019

Maple

m:=2; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # G. C. Greubel, Dec 03 2019

Mathematica

T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten

PARI

T(n,k) = my(m=2); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ G. C. Greubel, Dec 03 2019

Sage

def T(n, k):
    m=2
    if (k==0): return nth_prime(m)
    else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019

Formula

T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 2.

Extensions

Edited by G. C. Greubel, Dec 03 2019