A014148 -id:A014148 - cp's OEIS frontend

A122381 Numbers n such that A014148[n] is a prime.

1, 2, 3, 6, 10, 23, 31, 46, 55, 58, 66, 70, 82, 91, 118, 131, 151, 163, 182, 187, 198, 199, 203, 222, 275, 279, 334, 346, 351, 358, 402, 411, 462, 470, 515, 582, 591, 619, 639, 650, 667, 671, 679, 706, 739, 750, 767, 835, 851, 875, 882, 899, 919, 926, 962, 966

Offset: 1

Alexander Adamchuk, Aug 30 2006

nonn

Corresponding primes are listed in A122382[n] = A014148[ a(n) ] = {2,7,17,103,467,6577,17189,61627,109919,130531,198109,239579,399557,559313,...}.

			A014148[n] begins {2,7,17,34,62,103,161,238,338,467,627,824,1062,1343,...}.
a(1) = 1 because A014148[1] = 2 is prime.
a(2) = 2 because A014148[2] = 7 is prime.

Cf. A014148, A122382.

p=0;s=0;f=0;Do[p=Prime[n];s=s+p;f=f+s;If[PrimeQ[f],Print[{n,f}]],{n,1,2000}]
Position[Nest[Accumulate,Prime[Range[1000]],2],?PrimeQ]//Flatten (* _Harvey P. Dale, Jul 15 2023 *)

A014148[n] = Sum[ Sum[ Prime[k], {k,1,m} ], {m,1,n} ]. A014148[ a(n) ] = A122382[n].

A007504 Sum of the first n primes.

Original entry on oeis.org

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

A014285 a(n) = Sum_{j=1..n} j*prime(j).

Original entry on oeis.org

2, 8, 23, 51, 106, 184, 303, 455, 662, 952, 1293, 1737, 2270, 2872, 3577, 4425, 5428, 6526, 7799, 9219, 10752, 12490, 14399, 16535, 18960, 21586, 24367, 27363, 30524, 33914, 37851, 42043, 46564, 51290, 56505, 61941, 67750, 73944, 80457, 87377, 94716, 102318

Offset: 1

N. J. A. Sloane

Two consecutive terms cannot both be divisible by 4. - Tamas Sandor Nagy, Aug 04 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Partial sums of A033286. - Michel Marcus, Jun 18 2019

Cf. A007504, A014148.

[&+[k*NthPrime(k): k in [1..n]]: n in [1..40]];  // Bruno Berselli, Apr 30 2011

Join[{s=2},Table[s+=Prime[n]*n,{n,2,33}]] (* Vladimir Joseph Stephan Orlovsky, Dec 30 2010 *)
Accumulate[Table[i*Prime[i],{i,40}]] (* Harvey P. Dale, Sep 10 2014 *)

{a(n) = sum(j=1,n, j*prime(j))}; \\ G. C. Greubel, Jun 18 2019

[sum(j*nth_prime(j) for j in (1..n)) for n in (1..40)] # G. C. Greubel, Jun 18 2019

a(n) = n*A007504(n) - Sum_{k=1..n-1} A007504(k) = n*A007504(n) - A014148(n-1). - Pontus von Brömssen, Aug 29 2021

Offset changed to 1 and six terms added by Bruno Berselli, Apr 30 2011

A014150 Apply partial sum operator thrice to primes.

Original entry on oeis.org

2, 9, 26, 60, 122, 225, 386, 624, 962, 1429, 2056, 2880, 3942, 5285, 6956, 9008, 11500, 14493, 18054, 22254, 27166, 32869, 39446, 46986, 55586, 65347, 76372, 88768, 102644, 118113, 135302, 154342, 175370

Offset: 1

N. J. A. Sloane

nonn

Row 3 in A254858. - Reinhard Zumkeller, Feb 08 2015

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

Cf. A000040.
Cf. A007504, A014148, A178138, A254784.
Cf. A254858.

a014150 n = a014150_list !! (n-1)
a014150_list = (iterate (scanl1 (+)) a000040_list) !! 3
-- Reinhard Zumkeller, Feb 08 2015

lst={};s1=0;s2=0;s3=0;Do[s1=s1+Prime[n];s2=s2+s1;s3=s3+s2;AppendTo[lst, s3], {n, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 15 2008 *)
With[{nn=3},Nest[Accumulate[#]&,Prime[Range[50]],nn]] (* Harvey P. Dale, Feb 07 2015 *)

Offset fixed by Reinhard Zumkeller, Feb 08 2015

A254858 Iterated partial sums of prime numbers, square array read by diagonals.

Original entry on oeis.org

2, 2, 3, 2, 5, 5, 2, 7, 10, 7, 2, 9, 17, 17, 11, 2, 11, 26, 34, 28, 13, 2, 13, 37, 60, 62, 41, 17, 2, 15, 50, 97, 122, 103, 58, 19, 2, 17, 65, 147, 219, 225, 161, 77, 23, 2, 19, 82, 212, 366, 444, 386, 238, 100, 29, 2, 21, 101, 294, 578, 810, 830, 624, 338, 129, 31

Offset: 0

Reinhard Zumkeller, Feb 08 2015

Row n+1 = partial sums of row n.

T(n,1) = A002522(n+1); T(n,2) = A144396(n+1); T(n,3) = A002522(n+2).

			. n\k | 1  2  3   4   5    6    7     8     9    10    11     12     13
. ----+------------------------------------------------------------------
.  0  | 2  3  5   7  11   13   17    19    23    29    31     37     41 ..
.  1  | 2  5 10  17  28   41   58    77   100   129   160    197    238 ..
.  2  | 2  7 17  34  62  103  161   238   338   467   627    824   1062 ..
.  3  | 2  9 26  60 122  225  386   624   962  1429  2056   2880   3942 ..
.  4  | 2 11 37  97 219  444  830  1454  2416  3845  5901   8781  12723 ..
.  5  | 2 13 50 147 366  810 1640  3094  5510  9355 15256  24037  36760 ..
.  6  | 2 15 65 212 578 1388 3028  6122 11632 20987 36243  60280  97040 ..
.  7  | 2 17 82 294 872 2260 5288 11410 23042 44029 80272 140552 237592 ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..7259, first 120 diagonals.

Cf. A000040 (row 0), A007504 (row 1), A014148 (row 2), A014150 (row 3), A178138 (row 4), A254784 (row 5).
Cf. A007395 (column 1), A144396 (column 2), A002522 (column 3).
Cf. A125180 (antidiagonal sums), A125179 (diagonals downward).

a254858_tabl = diags [] $ iterate (scanl1 (+)) a000040_list where
   diags uss (vs:vss) = (map head wss) : diags (map tail wss) vss
                        where wss = vs : uss
a254858_list = concat a254858_tabl

nmax = 11;
row[0] = Prime[Range[nmax+1]];
row[n_] := row[n] = row[n-1] // Accumulate;
T[n_, k_] := row[n][[k]];
Table[T[n-k, k], {n, 0, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 11 2021 *)

A178138 Apply partial sum operator 4 times to primes.

Original entry on oeis.org

2, 11, 37, 97, 219, 444, 830, 1454, 2416, 3845, 5901, 8781, 12723, 18008, 24964, 33972, 45472, 59965, 78019, 100273, 127439, 160308, 199754, 246740, 302326, 367673, 444045, 532813, 635457, 753570, 888872, 1043214, 1218584

Offset: 1

Jonathan Vos Post, May 20 2010

Unlike the results of applying the partial sum operator once (A007504), twice (A014148), or thrice (A014150) to primes, this sequence begins with 4 primes. The next prime in the sequence is a(26) = 367673.

Row 4 in A254858. - Reinhard Zumkeller, Feb 08 2015

			a(15) = 2 + 9 + 26 + 60 + 122 + 225 + 386 + 624 + 962 + 1429 + 2056 + 2880 + 3942 + 5285 + 6956 = 24964 = 2^2 x 79^2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A007504, A014148, A014150, A178138, A254784.

Cf. A254858.

a178138 n = a178138_list !! (n-1)
a178138_list = (iterate (scanl1 (+)) a000040_list) !! 4
-- Reinhard Zumkeller, Feb 08 2015

Contribution from R. J. Mathar, Oct 19 2010: (Start)
A007504 := proc(n) option remember; add( ithprime(i),i=1..n) ; end proc:
A014148 := proc(n) option remember; add( A007504(i),i=1..n) ; end proc:
A014150 := proc(n) option remember; add( A014148(i),i=1..n) ; end proc:
A178138 := proc(n) option remember; add( A014150(i),i=1..n) ; end proc: seq(A178138(n),n=1..80) ; (End)

Nest[Accumulate[#]&,Prime[Range[40]],4] (* Harvey P. Dale, Sep 25 2014 *)

A254784 Apply partial sum operator 5 times to primes.

Original entry on oeis.org

2, 13, 50, 147, 366, 810, 1640, 3094, 5510, 9355, 15256, 24037, 36760, 54768, 79732, 113704, 159176, 219141, 297160, 397433, 524872, 685180, 884934, 1131674, 1434000, 1801673, 2245718, 2778531, 3413988, 4167558, 5056430, 6099644, 7318228, 8735337, 10376402

Offset: 1

Harvey P. Dale, Feb 07 2015

Row 5 in A254858. - Reinhard Zumkeller, Feb 08 2015

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007504, A014148, A014150, A178138.

Cf. A254858.

a254784 n = a254784_list !! (n-1)
a254784_list = (iterate (scanl1 (+)) a000040_list) !! 5
-- Reinhard Zumkeller, Feb 08 2015

With[{nn=5},Nest[Accumulate[#]&,Prime[Range[50]],nn]]

A157492 Apply partial sum operator twice to sequence of squares of the first n primes.

Original entry on oeis.org

4, 17, 55, 142, 350, 727, 1393, 2420, 3976, 6373, 9731, 14458, 20866, 29123, 39589, 52864, 69620, 90097, 115063, 145070, 180406, 221983, 270449, 326836, 392632, 468629, 555235, 653290, 763226, 885931, 1024765, 1180760, 1355524, 1549609

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001223, A014148, A014150, A061789, A069482, A129701.

Partial sums of A024450.

ListTools:-PartialSums(ListTools:-PartialSums([seq(ithprime(i)^2,i=1..100)])); # Robert Israel, May 14 2019

s0=s1=0;lst={};Do[p=Prime[n];s0+=p^2;s1+=s0;AppendTo[lst,s1],{n,5!}];lst
Nest[Accumulate,Prime[Range[40]]^2,2] (* Harvey P. Dale, Jan 01 2020 *)

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31

Offset: 1

Stefano Spezia, Jul 14 2019

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Wikipedia, Toeplitz matrix

Cf. A025581, A318173, A325516.

Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

[[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output

a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);

Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]

T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output

[[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

T(n, k) = A025581(n, k)*A000040(1 + k).

cp's OEIS Frontend

A122382 Primes of the form Sum[ Sum[ Prime[k], {k,1,m} ], {m,1,n} ] = A014148[n].

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A122381 Numbers n such that A014148[n] is a prime.

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A007504 Sum of the first n primes.

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A014285 a(n) = Sum_{j=1..n} j*prime(j).

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A014150 Apply partial sum operator thrice to primes.

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A254858 Iterated partial sums of prime numbers, square array read by diagonals.

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A178138 Apply partial sum operator 4 times to primes.

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