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n copies of n-th prime.
8

%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_