A037126 -id:A037126 - cp's OEIS frontend

A007504 Sum of the first n primes.

0, 2, 5, 10, 17, 28, 41, 58, 77, 100, 129, 160, 197, 238, 281, 328, 381, 440, 501, 568, 639, 712, 791, 874, 963, 1060, 1161, 1264, 1371, 1480, 1593, 1720, 1851, 1988, 2127, 2276, 2427, 2584, 2747, 2914, 3087, 3266, 3447, 3638, 3831, 4028, 4227, 4438, 4661, 4888

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

It appears that a(n)^2 - a(n-1)^2 = A034960(n). - Gary Detlefs, Dec 20 2011
This is true. Proof: By definition we have A034960(n) = Sum_{k = (a(n-1)+1)..a(n)} (2*k-1). Since Sum_{k = 1..n} (2*k-1) = n^2, it follows A034960(n) = a(n)^2 - a(n-1)^2, for n > 1. - Hieronymus Fischer, Sep 27 2012 [formulas above adjusted to changed offset of A034960 - Hieronymus Fischer, Oct 14 2012]
Row sums of the triangle in A037126. - Reinhard Zumkeller, Oct 01 2012
Ramanujan noticed the apparent identity between the prime parts partition numbers A000607 and the expansion of Sum_{k >= 0} x^a(k)/((1-x)...(1-x^k)), cf. A046676. See A192541 for the difference between the two. - M. F. Hasler, Mar 05 2014
For n > 0: row 1 in A254858. - Reinhard Zumkeller, Feb 08 2015
a(n) is the smallest number that can be partitioned into n distinct primes. - Alonso del Arte, May 30 2017
For a(n) < m < a(n+1), n > 0, at least one m is a perfect square.
Proof: For n = 1, 2, ..., 6, the proposition is clear. For n > 6, a(n) < ((prime(n) - 1)/2)^2, set (k - 1)^2 <= a(n) < k^2 < ((prime(n) + 1)/2)^2, then k^2 < (k - 1)^2 + prime(n) <= a(n) + prime(n) = a(n+1), so m = k^2 is this perfect square. - Jinyuan Wang, Oct 04 2018
For n >= 5 we have a(n) < ((prime(n)+1)/2)^2. This can be shown by noting that ((prime(n)+1)/2)^2 - ((prime(n-1)+1)/2)^2 - prime(n) = (prime(n)+prime(n-1))*(prime(n)-prime(n-1)-2)/4 >= 0. - Jianing Song, Nov 13 2022
Washington gives an oscillation formula for |a(n) - pi(n^2)|, see links. - Charles R Greathouse IV, Dec 07 2022

E. Bach and J. Shallit, §2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms, MIT Press, Cambridge, MA, 1996.
H. L. Nelson, "Prime Sums", J. Rec. Math., 14 (1981), 205-206.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 0..100000
C. Axler, On a Sequence involving Prime Numbers, J. Int. Seq. 18 (2015) # 15.7.6.
Christian Axler, New bounds for the sum of the first n prime numbers, arXiv:1606.06874 [math.NT], 2016.
P. Hecht, Post-Quantum Cryptography: S_381 Cyclic Subgroup of High Order, International Journal of Advanced Engineering Research and Science (IJAERS, 2017) Vol. 4, Issue 6, 78-86.
R. J. Mathar, Table of 100000n, a(100000n) for n = 1..10000
Romeo Meštrović, Curious conjectures on the distribution of primes among the sums of the first 2n primes, arXiv:1804.04198 [math.NT], 2018.
Vladimir Shevelev, Asymptotics of sum of the first n primes with a remainder term
Nilotpal Kanti Sinha, On the asymptotic expansion of the sum of the first n primes, arXiv:1011.1667 [math.NT], 2010-2015.
Lawrence C. Washington, Sums of Powers of Primes II, arXiv preprint (2022). arXiv:2209.12845 [math.NT]
Eric Weisstein's World of Mathematics, Prime Sums
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A000041, A034386, A111287, A013916, A013918 (primes), A045345, A050247, A050248, A068873, A073619, A034387, A014148, A014150, A178138, A254784, A254858.
See A122989 for the value of Sum_{n >= 1} 1/a(n).
Cf. A008472, A002110, A038107.

P:=Filtered([1..250],IsPrime);;
a:=Concatenation([0],List([1..Length(P)],i->Sum([1..i],k->P[k]))); # Muniru A Asiru, Oct 07 2018

a007504 n = a007504_list !! n
a007504_list = scanl (+) 0 a000040_list
-- Reinhard Zumkeller, Oct 01 2014, Oct 03 2011

[0] cat [&+[ NthPrime(k): k in [1..n]]: n in [1..50]]; // Bruno Berselli, Apr 11 2011 (adapted by Vincenzo Librandi, Nov 27 2015 after Hasler's change on Mar 05 2014)

s1:=[2]; for n from 2 to 1000 do s1:=[op(s1),s1[n-1]+ithprime(n)]; od: s1;
A007504 := proc(n)
    add(ithprime(i), i=1..n) ;
end proc: # R. J. Mathar, Sep 20 2015

Accumulate[Prime[Range[100]]] (* Zak Seidov, Apr 10 2011 *)
primeRunSum = 0; Table[primeRunSum = primeRunSum + Prime[k], {k, 100}] (* Zak Seidov, Apr 16 2011 *)

A007504(n) = sum(k=1,n,prime(k)) \\ Michael B. Porter, Feb 26 2010

a(n) = vecsum(primes(n)); \\ Michel Marcus, Feb 06 2021

from itertools import accumulate, count, islice
from sympy import prime
def A007504_gen(): return accumulate(prime(n) if n > 0 else 0 for n in count(0))
A007504_list = list(islice(A007504_gen(),20)) # Chai Wah Wu, Feb 23 2022

a(n) ~ n^2 * log(n) / 2. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 24 2001 (see Bach & Shallit 1996)
a(n) = A014284(n+1) - 1. - Jaroslav Krizek, Aug 19 2009
a(n+1) - a(n) = A000040(n+1). - Jaroslav Krizek, Aug 19 2009
a(A051838(n)) = A002110(A051838(n)) / A116536(n). - Reinhard Zumkeller, Oct 03 2011
a(n) = min(A068873(n), A073619(n)) for n > 1. - Jonathan Sondow, Jul 10 2012
a(n) = A033286(n) - A152535(n). - Omar E. Pol, Aug 09 2012
For n >= 3, a(n) >= (n-1)^2 * (log(n-1) - 1/2)/2 and a(n) <= n*(n+1)*(log(n) + log(log(n))+ 1)/2. Thus a(n) = n^2 * log(n) / 2 + O(n^2*log(log(n))). It is more precise than in Fares's comment. - Vladimir Shevelev, Aug 01 2013
a(n) = (n^2/2)*(log n + log log n - 3/2 + (log log n - 3)/log n + (2 (log log n)^2 - 14 log log n + 27)/(4 log^2 n) + O((log log n/log n)^3)) [Sinha]. - Charles R Greathouse IV, Jun 11 2015
G.f: (x*b(x))/(1-x), where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 10 2016
a(n) = A008472(A002110(n)), for n > 0. - Michel Marcus, Jul 16 2020

More terms from Stefan Steinerberger, Apr 11 2006

a(0) = 0 prepended by M. F. Hasler, Mar 05 2014

A104887 Triangle T(n,k) = (n-k+1)-th prime, read by rows.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 29 2005

Repeatedly writing the prime sequence backwards.

Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A104887 is the reverse reluctant sequence of sequence the prime numbers (A000040). - Boris Putievskiy, Dec 13 2012

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

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Reflected triangle of A037126.

Cf. A098012 (partial products per row).

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

import Data.List (inits)
a104887 n k = a104887_tabl !! (n-1) !! (k-1)
a104887_row n = a104887_tabl !! (n-1)
a104887_tabl = map reverse $ tail $ inits a000040_list
-- Reinhard Zumkeller, Oct 02 2014

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

Module[{nn=15,prms},prms=Prime[Range[nn]];Table[Reverse[Take[prms,n]],{n,nn}]]//Flatten (* Harvey P. Dale, Aug 10 2021 *)

T(n,k) = A000040(n-k+1); a(n) = A000040(A004736(n)).

a(n) = A000040(m), where m=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012

Edited by Ralf Stephan, Apr 05 2009

A220280 The reluctant sequence for A002260.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3

Offset: 1

Boris Putievskiy, Dec 12 2012

The reluctant sequence B for a sequence A is a triangular array in which row k (>= 1) consists of the first k terms of A.

Here A002260 is the reluctant sequence for the sequence 1,2,3,... of positive numbers (A000027).

			A002260 begins
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, ...
so the first few rows of the new triangle are
   1,
   1, 1,
   1, 1, 2,
   1, 1, 2, 1,
   1, 1, 2, 1, 2,
   1, 1, 2, 1, 2, 3,
   ...
                                                                               ~

Boris Putievskiy, Table of n, a(n) for n = 1..10000
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A002260, A002262, A037126, A059268, A215026.

t=int((math.sqrt(8*n-7) - 1)/ 2)
n1=n-t*(t+1)/2
t1=int((math.sqrt(8*n1-7) - 1)/ 2)
a=n1-t1*(t1+1)/2

a(n) = n1 - t1(t1+1)/2, where n1 = n - t(t+1)/2, t1 = floor[(-1+sqrt(8*n1-7))/2], t=floor[(-1+sqrt(8*n-7))/2]. For example, a(6)=2 since t=2, t1=1, n1=3.

Edited by N. J. A. Sloane, Jun 07 2024

A109400 For all k: the first k numbers followed by the first k primes.

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 5, 1, 2, 3, 4, 2, 3, 5, 7, 1, 2, 3, 4, 5, 2, 3, 5, 7, 11, 1, 2, 3, 4, 5, 6, 2, 3, 5, 7, 11, 13, 1, 2, 3, 4, 5, 6, 7, 2, 3, 5, 7, 11, 13, 17, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 5, 7, 11, 13, 17, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 5, 7, 11, 13, 17, 19, 23

Offset: 1

Reinhard Zumkeller, Jun 27 2005

nonn

a(k*(k-1) + m) = a(A002378(k-1) + m) = m for 1<=m<=k;
a(k^2 + m) = a(A000290(k) + m) = A000040(m) for 1<=m<=k;
see A109401 for number of primes and A109402 for partial sums.

			1, 2; 1 2, 2 3; 1 2 3, 2 3 5; 1 2 3 4, 2 3 5 7; 1 2 ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002260, A037126, A082500, A000040.

a109400 n = a109400_list !! (n-1)
a109400_list = concat $ zipWith (++) a002260_tabl a037126_tabl
-- Reinhard Zumkeller, Jun 23 2015, Dec 11 2011

With[{prs=Prime[Range[20]]},Flatten[Table[{Range[n],Take[prs,n]},{n,10}]]] (* Harvey P. Dale, Dec 08 2011 *)

A138143 Triangle read by rows in which row n lists p(1), p(2), ..., p(n), p(n-1), ..., p(1), where p(i) = i-th prime.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 09 2008

Row n contains 2n-1 terms and each column lists the prime numbers A000040.

Triangle of primes mentioned in A061802.

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

Cf. A000040, A061802, A066622, A037126.

nn=10;Module[{pr=Prime[Range[nn]],e},Flatten[Table[e=Take[pr,n];Join[ e,Reverse[Most[e]]],{n,nn}]]] (* Harvey P. Dale, Mar 14 2015 *)

Edited by N. J. A. Sloane, Apr 07 2008, Nov 15 2008

A264662 Triangle read by rows: row n contains the first n primes in lexicographical order of their mirrored binary representation.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 20 2015

T(n,A263856(n)) = A000040(n): A263856(n) = index of prime(n) in n-th row.

			.   n |   T(n,k), k=1..n
. ----+-----------------------------------------------------------------
.   1 | 2                          01
.   2 | 2  3                       01 11
.   3 | 2  5  3                    01 101   11
.   4 | 2  5  3  7                 01 101   11   111
.   5 | 2  5  3 11  7              01 101   11   1101 111
.   6 | 2  5 13  3 11  7           01 101   1011 11   1101 111
.   7 | 2 17  5 13  3 11  7        01 10001 101  1011 11   1101  111
.   8 | 2 17  5 13  3 19 11  7     01 10001 101  1011 11   11001 1101 111
.   9 | 2 17  5 13  3 19 11  7 23
.  10 | 2 17  5 13 29  3 19 11  7 23
.  11 | 2 17  5 13 29  3 19 11  7 23 31
.  12 | 2 17  5 37 13 29  3 19 11  7 23 31
.  13 | 2 17 41  5 37 13 29  3 19 11  7 23 31
.  14 | 2 17 41  5 37 13 29  3 19 11 43 7  23 31
.  15 | 2 17 41  5 37 13 29  3 19 11 43  7 23 47 31
.  16 | 2 17 41  5 37 53 13 29  3 19 11 43  7 23 47 31
.  17 | 2 17 41  5 37 53 13 29  3 19 11 43 59  7 23 47 31
.  18 | 2 17 41  5 37 53 13 29 61  3 19 11 43 59  7 23 47 31
.  19 | 2 17 41  5 37 53 13 29 61  3 67 19 11 43 59  7 23 47 31
.  20 | 2 17 41  5 37 53 13 29 61  3 67 19 11 43 59  7 71 23 47 31

Reinhard Zumkeller, Rows n = 1..250 of triangle, flattened

Cf. A263846, A000040, A007088, A007504 (row sums), A264666 (partial row products), A037126 (rows sorted naturally).

import Data.List (inits, sortBy); import Data.Function (on)
a264662 n k = a264662_tabl !! (n-1) !! (n-1)
a264662_row n = a264662_tabl !! (n-1)
a264662_tabl = map (sortBy (compare `on` (reverse . show . a007088))) $
                   tail $ inits a000040_list

row[n_] := SortBy[Prime[Range[n]], StringJoin[ToString /@ Reverse[IntegerDigits[#, 2]]]&];
Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Sep 25 2021 *)

A332644 Largest of the least integers of prime signatures over all partitions of n into distinct parts.

Original entry on oeis.org

1, 2, 4, 12, 24, 72, 360, 720, 2160, 10800, 75600, 151200, 453600, 2268000, 15876000, 174636000, 349272000, 1047816000, 5239080000, 36673560000, 403409160000, 5244319080000, 10488638160000, 31465914480000, 157329572400000, 1101307006800000, 12114377074800000

Offset: 0

Alois P. Heinz, Feb 18 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..712
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Subsequence of A025487, A055932, A087980.

Cf. A000009, A000040, A000217, A002110, A002260, A003056, A001221, A001222, A007814, A037126, A046523, A123578, A328524.

b:= proc(n, i, j) option remember;
      `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*
      ithprime(n-(t-> t*(t+1)/2)(floor((sqrt(8*n-7)-1)/2))))
    end:
seq(a(n), n=0..30);

b[n_, i_, j_] := b[n, i, j] = If[i(i+1)/2 < n, 0, If[n == 0, 1, Max[b[n, i - 1, j], Prime[j]^i b[n - i, Min[n - i, i - 1], j + 1]]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 30] (* Jean-François Alcover, May 07 2020, after 1st Maple program *)

a(n) = A328524(n,A000009(n)).
A001221(a(n)) = A003056(n).
A001222(a(n)) = n.
A046523(a(n)) = a(n).
a(n)/a(n-1) = A037126(n) = A000040(n-A000217(A003056(n))) for n > 0.
a(n) in { A025487 }.
a(n) in { A055932 }.
a(n) in { A087980 }.
A007814(a(n)) = A123578(n).

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 *)

A354271 Irregular array of the prime numbers read by rows.

Original entry on oeis.org

2, 3, 2, 5, 7, 3, 11, 2, 5, 13, 17, 3, 11, 19, 2, 7, 23, 17, 29, 5, 13, 31, 7, 37, 3, 11, 19, 29, 41, 2, 5, 13, 43, 17, 47, 3, 11, 23, 41, 53, 2, 7, 37, 59, 31, 61, 5, 13, 23, 43, 67, 7, 19, 29, 37, 59, 71, 3, 11, 41, 73, 2, 5, 17, 47, 79, 19, 31, 53, 71, 83, 3, 67

Offset: 1

Tamas Sandor Nagy, May 22 2022

The construction of the array is made in an orthogonal grid with columns and rows.
Along the sloping upper boundary of the array are written the guiding prime numbers, each in a column of its value and in a row of its index. From these leading entries, on downward and leftward running antidiagonal lines the preceding primes are entered. In reverse order and with the due prime gaps, these will fall into the columns of their own value, below the guiding primes on top.
The antidiagonals are the same as the rows of the triangle in A037126.
The rows that begin with 2's end with A256491.
Row n lists all primes of the form A000040(n - k) - k for positive k. - Thomas Scheuerle, May 23 2022

			.  2
.  .  3
.  2  .  .  5
.  .  .  .  .  .  7
.  .  3  .  .  .  .  .  .  . 11
.  2  .  .  5  .  .  .  .  .  .  . 13
.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 17
.  .  3  .  .  .  .  .  .  . 11  .  .  .  .  .  .  . 19
.  2  .  .  .  .  7  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 23
.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 17  .  .  .  .  .  .
.  .  .  .  5  .  .  .  .  .  .  . 13  .  .  .  .  .  .  .  .  .  .
.  .  .  .  .  .  7  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
.  .  3  .  .  .  .  .  .  . 11  .  .  .  .  .  .  . 19  .  .  .  .
.  2  .  .  5  .  .  .  .  .  .  . 13  .  .  .  .  .  .  .  .  .  .
.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 17  .  .  .  .  .  .
.  .  3  .  .  .  .  .  .  . 11  .  .  .  .  .  .  .  .  .  .  . 23
.  2  .  .  .  .  7  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

Michael De Vlieger, Table of n, a(n) for n = 1..10701 (rows n = 1..400, flattened)
Michael De Vlieger, Bitmap of 2^(pi(T(n,k)) - 1) for n = 1..1024.
Thomas Scheuerle, The first 1000 rows, a(1 .. 58521) as scatter plot.
Thomas Scheuerle, Lengths of the first 10000 rows as plot against row number.

Cf. A000040, A037126, A256491.

function a = A354271( max_row )
    p = primes(max_row*floor(2*max_row*log(max_row)));
    a = [];
    for r = 1:max_row
        row = p(1:r)-(r-1:-1:0);
        row = row(isprime(max(row,0)) > 0);
        a = [a row];
    end
end % Thomas Scheuerle, May 23 2022

Table[Select[Array[Prime[#] - (n - #) &, n], And[# > 0, PrimeQ[#]] &], {n, 24}] // Flatten (* Michael De Vlieger, May 25 2022 *)
(* Extract data from the bitmap: set k to number of rows desired, up to 1024 *)
k = 120; Map[Prime /@ Position[#, 0.][[All, 1]] &, ImageData[Import["https://oeis.org/A354271/a354271_2.png"]][[1 ;; k]] ] // Flatten (* Michael De Vlieger, May 25 2022 *)

cp's OEIS Frontend

A007504 Sum of the first n primes.

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Magma

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PARI

PARI

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A104887 Triangle T(n,k) = (n-k+1)-th prime, read by rows.

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A220280 The reluctant sequence for A002260.

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A109400 For all k: the first k numbers followed by the first k primes.

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A138143 Triangle read by rows in which row n lists p(1), p(2), ..., p(n), p(n-1), ..., p(1), where p(i) = i-th prime.

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A264662 Triangle read by rows: row n contains the first n primes in lexicographical order of their mirrored binary representation.

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A332644 Largest of the least integers of prime signatures over all partitions of n into distinct parts.