A155151 - OEIS

%I #68 Sep 08 2022 08:45:40

%S 10,16,26,22,36,50,28,46,64,82,34,56,78,100,122,40,66,92,118,144,170,

%T 46,76,106,136,166,196,226,52,86,120,154,188,222,256,290,58,96,134,

%U 172,210,248,286,324,362,64,106,148,190,232,274,316,358,400,442,70,116,162

%N Triangle T(n, k) = 4*n*k + 2*n + 2*k + 2, read by rows.

%C First column: A016957, second column: A017341, third column: 2*A017029, fourth column: A082286. - _Vincenzo Librandi_, Nov 21 2012

%C Conjecture: Let p = prime number. If 2^p belongs to the sequence, then 2^p-1 is not a Mersenne prime. - _Vincenzo Librandi_, Dec 12 2012

%C Conjecture is true because if T(n, k) = 2^p with p prime, then 2^p-1 = 4*n*k + 2*n + 2*k + 1 = (2*n+1)*(2*k+1) hence 2^p-1 is not prime. - _Michel Marcus_, May 31 2015

%C It appears that T(m,p) = 2^p for Lucasian primes (A002515) greater than 3. For instance: T(44, 11) = 2^11, T(89240, 23) = 2^23. - _Michel Marcus_, May 28 2015

%C For n > 1, ascending numbers along the diagonal are also terms of the even principal diagonal of a 2n X 2n spiral (A137928). - _Avi Friedlich_, May 21 2015

%H Vincenzo Librandi, <a href="/A155151/b155151.txt">Rows n = 1..100, flattened</a>

%F T(n, k) = 2*A144650(n, k).

%F Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n + 3) = n*A014105(n+2) =

%e Triangle begins

%e   10;

%e   16,  26;

%e   22,  36,  50;

%e   28,  46,  64,  82;

%e   34,  56,  78, 100, 122;

%e   40,  66,  92, 118, 144, 170;

%e   46,  76, 106, 136, 166, 196, 226;

%e   52,  86, 120, 154, 188, 222, 256, 290;

%e   58,  96, 134, 172, 210, 248, 286, 324, 362;

%e   64, 106, 148, 190, 232, 274, 316, 358, 400, 442;

%p seq(seq( 2*(2*n*k+n+k+1), k=1..n), n=1..15) # _G. C. Greubel_, Mar 21 2021

%t T[n_,k_]:=4*n*k + 2*n + 2*k + 2; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* _Vincenzo Librandi_, Nov 21 2012 *)

%o (Magma) [4*n*k + 2*n + 2*k + 2: k in [1..n], n in [1..11]]; // _Vincenzo Librandi_, Nov 21 2012

%o (Sage) flatten([[2*(2*n*k+n+k+1) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 21 2021

%Y Cf. A000043, A000668, A016957, A017029, A017341, A054723, A082286, A144650.

%K nonn,tabl,easy

%O 1,1

%A _Vincenzo Librandi_, Jan 21 2009

%E Edited by _Robert Hochberg_, Jun 21 2010