A138143 -id:A138143 - cp's OEIS frontend

A037126 Triangle T(n,k) = prime(k) for k = 1..n.

2, 2, 3, 2, 3, 5, 2, 3, 5, 7, 2, 3, 5, 7, 11, 2, 3, 5, 7, 11, 13, 2, 3, 5, 7, 11, 13, 17, 2, 3, 5, 7, 11, 13, 17, 19, 2, 3, 5, 7, 11, 13, 17, 19, 23, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 2, 3, 5, 7, 11, 13, 17

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

Or, triangle read by rows in which row n lists first n primes.

Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A037126 is reluctant sequence of the prime numbers A000040. - Boris Putievskiy, Dec 12 2012

			Triangle begins:
..... 2
.... 2,3
... 2,3,5
.. 2,3,5,7
. 2,3,5,7,11
...

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A000040, A002260, A037126, A138139, A138140, A138143.

Cf. A007504 (row sums).

P:=Filtered([1..200],IsPrime);;
T:=Flat(List([1..13],n->List([1..n],k->P[k]))); # Muniru A Asiru, Mar 16 2019

a037126 n k = a037126_tabl !! (n-1) !! (k-1)
a037126_row n = a037126_tabl !! (n-1)
a037126_tabl = map (`take` a000040_list) [1..]
-- Reinhard Zumkeller, Oct 01 2012

T:=(n,k)->ithprime(k): seq(seq(T(n,k),k=1..n),n=1..13); # Muniru A Asiru, Mar 16 2019

Flatten[ Table[ Prime[ i], {n, 12}, {i, n}]] (* Robert G. Wilson v, Aug 18 2005 *)
Module[{nn=15,prs},prs=Prime[Range[nn]];Table[Take[prs,n],{n,nn}]]// Flatten (* Harvey P. Dale, May 02 2017 *)

As a linear array, the sequence is a(n) = A000040(m), where m = n-t(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 12 2012

A061802 Sum of n-th row of triangle of primes: 2; 2 3 2; 2 3 5 3 2; 2 3 5 7 5 3 2; ...; where n-th row contains 2n+1 terms.

Original entry on oeis.org

2, 7, 15, 27, 45, 69, 99, 135, 177, 229, 289, 357, 435, 519, 609, 709, 821, 941, 1069, 1207, 1351, 1503, 1665, 1837, 2023, 2221, 2425, 2635, 2851, 3073, 3313, 3571, 3839, 4115, 4403, 4703, 5011, 5331, 5661, 6001, 6353, 6713, 7085, 7469, 7859, 8255, 8665

Offset: 0

Amarnath Murthy, May 28 2001

Row sums of A138143. - Omar E. Pol, Feb 13 2014

For n = 3..9, a(n) = 3*(n^2 - 3*n + 5). - Nicholas Drozd, Apr 10 2021

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Cf. A001043 (first differences), A007504, A138143.

Partial sums of A011974.

Accumulate[Join[{2},ListConvolve[{1,1},Prime[Range[100]]]]] (* Paolo Xausa, Oct 31 2023 *)

{ n=-1; a=q=0; forprime (p=2, prime(1001), write("b061802.txt", n++, " ", a+=p + q); q=p ) } \\ Harry J. Smith, Jul 28 2009

a(n) = a(n-1) + prime(n) + prime(n-1).

a(n) = A007504(n) + A007504(n+1) so we have the asymptotic expansion a(n) ~ n^2*log(n). - Henry Bottomley, May 30 2001

More terms from Larry Reeves (larryr(AT)acm.org), May 29 2001

A138117 Triangle read by rows: row n lists the first 2n-1 prime numbers.

Original entry on oeis.org

2, 2, 3, 5, 2, 3, 5, 7, 11, 2, 3, 5, 7, 11, 13, 17, 2, 3, 5, 7, 11, 13, 17, 19, 23, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Offset: 1

Omar E. Pol, Mar 14 2008, corrected Mar 15 2008

			Triangle begins:
....... 2
..... 2,3,5
... 2,3,5,7,11
. 2,3,5,7,11,13,17

Cf. A000040, A037126, A138139, A138143.

nn=10;Flatten[Table[Take[Prime[Range[2nn+1]],2n+1],{n,0,nn}]] (* Harvey P. Dale, Aug 16 2011 *)

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A037126 Triangle T(n,k) = prime(k) for k = 1..n.

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A061802 Sum of n-th row of triangle of primes: 2; 2 3 2; 2 3 5 3 2; 2 3 5 7 5 3 2; ...; where n-th row contains 2n+1 terms.

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A138117 Triangle read by rows: row n lists the first 2n-1 prime numbers.

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