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%I #71 Feb 29 2024 19:10:12
%S 2,3,3,5,5,5,7,7,7,7,11,11,11,11,11,13,13,13,13,13,13,17,17,17,17,17,
%T 17,17,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,23,29,29,29,29,
%U 29,29,29,29,29,29,31,31,31,31,31,31,31,31,31,31,31
%N n copies of n-th prime.
%C Seen as a triangle read by rows: T(n,k) = A000040(n), 1 <= k <= n; row sums = A033286; central terms = A031368. - _Reinhard Zumkeller_, Aug 05 2009
%C Seen as a square array read by antidiagonals, a subtable of the binary operation multiplication tables A297845, A306697 and A329329. - _Peter Munn_, Jan 15 2020
%D Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought", Basic Books, 1995.
%H Reinhard Zumkeller, <a href="/A005145/b005145.txt">Rows n = 1..125 of triangle, flattened</a>
%F From Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006: (Start)
%F a(n) = prime(floor(1/2 + sqrt(2*n))).
%F a(n) = A000040(A002024(n)). (End)
%F From _Peter Munn_, Jan 15 2020: (Start)
%F When viewed as a square array A(n,k), the following hold for n >= 1, k >= 1:
%F A(n,k) = prime(n+k-1).
%F A(n,1) = A(1,n) = prime(n), where prime(n) = A000040(n).
%F A(n+1,k) = A(n,k+1) = A003961(A(n,k)).
%F A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)) = A329329(A(n,1), A(1,k)).
%F (End)
%F Sum_{n>=1} 1/a(n)^2 = A097906. - _Amiram Eldar_, Aug 16 2022
%e Triangle begins:
%e 2;
%e 3, 3;
%e 5, 5, 5;
%e 7, 7, 7, 7;
%e ...
%t Table[Prime[Floor[1/2 + Sqrt[2*n]]], {n, 1, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
%t Flatten[Table[Table[Prime[n], {n}], {n, 12}]] (* _Alonso del Arte_, Jan 18 2012 *)
%t Table[PadRight[{},n,Prime[n]],{n,15}]//Flatten (* _Harvey P. Dale_, Feb 29 2024 *)
%o (Haskell)
%o a005145 n k = a005145_tabl !! (n-1) !! (k-1)
%o a005145_row n = a005145_tabl !! (n-1)
%o a005145_tabl = zipWith ($) (map replicate [1..]) a000040_list
%o a005145_list = concat a005145_tabl
%o -- _Reinhard Zumkeller_, Jul 12 2014, Mar 18 2011, Oct 17 2010
%o (PARI) a(n) = prime(round(sqrt(2*n))) \\ _Charles R Greathouse IV_, Oct 23 2015
%o (Magma) [NthPrime(Round(Sqrt(2*n))): n in [1..60]]; // _Vincenzo Librandi_, Jan 18 2020
%o (Python)
%o from sympy import primerange
%o a = []; [a.extend([pn]*n) for n, pn in enumerate(primerange(1, 32), 1)]
%o print(a) # _Michael S. Branicky_, Jul 13 2022
%Y Sequences with similar definitions: A002024, A175944.
%Y Cf. A000040 (range of values), A003961, A031368 (main diagonal), A033286 (row sums), A097906.
%Y Subtable of A297845, A306697, A329329.
%K nonn,nice,tabl
%O 1,1
%A _Russ Cox_
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