A005145 n copies of n-th prime.

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, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31

Offset

1,1

Comments

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

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

References

Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought", Basic Books, 1995.

Links

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Formula

From Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006: (Start)

a(n) = prime(floor(1/2 + sqrt(2*n))).

a(n) = A000040(A002024(n)). (End)

From Peter Munn, Jan 15 2020: (Start)

When viewed as a square array A(n,k), the following hold for n >= 1, k >= 1:

A(n,k) = prime(n+k-1).

A(n,1) = A(1,n) = prime(n), where prime(n) = A000040(n).

A(n+1,k) = A(n,k+1) = A003961(A(n,k)).

A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)) = A329329(A(n,1), A(1,k)).

(End)

Sum_{n>=1} 1/a(n)^2 = A097906. - Amiram Eldar, Aug 16 2022

Example

Triangle begins:
  2;
  3, 3;
  5, 5, 5;
  7, 7, 7, 7;
  ...

Mathematica

Table[Prime[Floor[1/2 + Sqrt[2*n]]], {n, 1, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
Flatten[Table[Table[Prime[n], {n}], {n, 12}]] (* Alonso del Arte, Jan 18 2012 *)
Table[PadRight[{}, n, Prime[n]], {n, 15}]//Flatten (* Harvey P. Dale, Feb 29 2024 *)

Prog

(Haskell)
a005145 n k = a005145_tabl !! (n-1) !! (k-1)
a005145_row n = a005145_tabl !! (n-1)
a005145_tabl = zipWith ($) (map replicate [1..]) a000040_list
a005145_list = concat a005145_tabl
-- Reinhard Zumkeller, Jul 12 2014, Mar 18 2011, Oct 17 2010
(PARI) a(n) = prime(round(sqrt(2*n))) \\ Charles R Greathouse IV, Oct 23 2015
(Magma) [NthPrime(Round(Sqrt(2*n))): n in [1..60]]; // Vincenzo Librandi, Jan 18 2020
(Python)
from sympy import primerange
a = []; [a.extend([pn]*n) for n, pn in enumerate(primerange(1, 32), 1)]
print(a) # Michael S. Branicky, Jul 13 2022

Crossrefs

Sequences with similar definitions: A002024, A175944.

Cf. A000040 (range of values), A003961, A031368 (main diagonal), A033286 (row sums), A097906.

Subtable of A297845, A306697, A329329.

Sequence in context: A053046 A261179 A066658 * A280740 A156350 A076367

Adjacent sequences: A005142 A005143 A005144 * A005146 A005147 A005148

Keyword

nonn,nice,tabl

Author

Russ Cox

Status

approved