A068873 -id:A068873 - cp's OEIS frontend

A102330 Triangle read by rows: n-th row consists of the lexicographically earliest set of n distinct primes whose sum is A068873(n).

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

Offset: 1

Giovanni Teofilatto, Jan 30 2005

			Triangle begins:
  2
  2,3
  3,5,11
  2,3,5,7
  3,5,7,11,17
  2,3,5,7,11,13
  3,5,7,11,13,17,23
  2,3,5,7,11,13,19,23
  3,5,7,11,13,17,19,23,29
  2,3,5,7,11,13,17,19,23,31
  3,5,7,11,13,17,19,23,29,31,41
  2,3,5,7,11,13,17,19,23,29,31,37
  3,5,7,11,13,17,19,23,29,31,37,41,47
  2,3,5,7,11,13,17,19,23,29,31,37,41,43
  3,5,7,11,13,17,19,23,29,31,37,41,43,47,53

David A. Corneth, Table of n, a(n) for n = 1..10011 (first 141 rows flattened, first 300 terms from Jean-Francois Alcover, terms 301..2485 from Robert Israel)
Robert Israel, Table of n, a(n) for n = 1..2485 (rows 1 to 70, flattened), replacing incorrect b-file from Jean-Francois Alcover

By definition, row sums are A068873.

g:= proc(n,k,m) option remember; # lex earliest set of k distinct primes > m with sum n
   local q,v;
  if k = 1 then
    if isprime(n) and n > m then return [n] else return NULL fi
  fi;
  q:= m;
  do
    q:= nextprime(q);
    if n < k*q then return NULL fi;
    v:= procname(n-q,k-1,q);
    if v <> NULL then return [q,op(v)] fi
  od
end proc:
f:= proc(k)
local p,i,v;
p:= add(ithprime(i),i=1..k)-1;
do
  p:= nextprime(p);
  v:= g(p,k,0);
  if v <> NULL then return v fi
od
end proc:
for k from 1 to 30 do
  f(k)
od; # Robert Israel, May 12 2025

(* Computation verified with A068873. *)
row[n_] := Module[{s, m}, s = Select[{#, Total[#]}& /@ Subsets[ Prime[ Range[n+4]], {n}], PrimeQ[#[[2]]]&]; m = MinimalBy[s, #[[2]]&, 1]; If[s != {}, Return[m[[1, 1]]]]];
Array[row, 49] // Flatten (* Jean-François Alcover, Apr 23 2020 *)

Edited, corrected and extended by Ray Chandler, Feb 02 2005

Edited by N. J. A. Sloane, May 07 2014

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

A073619 a(1) = 0; a(n) = smallest composite number which is a sum of n distinct primes.

Original entry on oeis.org

0, 8, 10, 21, 28, 45, 58, 77, 100, 129, 160, 201, 238, 285, 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: 1

Amarnath Murthy, Aug 07 2002

nonn

			a(4) = 21 as 21 = 2+3+5+11 is the smallest composite number expressible as sum of four distinct primes.

Jean-François Alcover, Conjectured terms up to a(200)

Cf. A068873, A073620, A073621.

Min(a(n), A068873(n)) = A007504(n) for n > 1. - Jonathan Sondow, Jul 10 2012

More terms from Sascha Kurz, Feb 03 2003

A383725 a(n) is the least number k such that omega(k) = n and the largest prime factor of k equals the sum of its remaining prime factors, where omega(k) = A001221(k).

Original entry on oeis.org

30, 3135, 3570, 844305, 1231230, 463798335, 1089218130, 410825520105, 905980145070, 818186519485335, 1461885412557570, 2023416377587710105, 3676255934199278430, 6175645531427513476335, 14590719651042312667890, 29263451149172039260325865, 67794672364404337821058590

Offset: 3

Paolo Xausa, May 07 2025

nonn

a(n) and n have opposite parity. - David A. Corneth, May 08 2025

			a(3) = 30 is the smallest number having 3 distinct prime factors (namely 2, 3, and 5) such that the largest one is the sum of the others (2 + 3 = 5).
a(4) = 3135 is the smallest number having 4 distinct prime factors (namely 3, 5, 11 and 19) such that the largest one is the sum of the others (3 + 5 + 11 = 19).

Cf. A001221, A002110, A068873, A102330, A365795, A382469, A383728, A383729.

First column of A383726.

isok(k, n) = my(f=factor(k)); (omega(f)==n) && (vecsum(f[,1]) == 2*vecmax(f[,1]));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, May 08 2025

a(n) = (Product_{i=1..n-1} A102330(n-1,i))*A068873(n-1).
a(n) = A383726(n,1).
a(n) >= A002110(n).

A286263 The smallest weight possible for a prime vector of order n.

Original entry on oeis.org

2, 8, 19, 26, 43, 56, 79, 104, 127, 166, 223, 258, 307, 348

Offset: 1

Dmitry Kamenetsky, May 05 2017

A prime vector of order n is an array of n distinct primes P = (p_1, p_2, ..., p_n), such that every sum of an odd number of consecutive elements is also prime. The weight of the prime vector is the sum of its elements. For full details see Kamenetsky's paper.
Calculations by Kamenetsky and J. K. Andersen show that a(15-17) are likely to be 443, 522 and 641.
Calculations by J. K. Andersen show that a(18-21) are likely to be 762, 881, 1002 and 1259.
J. K. Andersen found the best upper bounds for a(22-23) as 1716 and 1931.
For odd n, a(n) <= A068873(n) (smallest prime which is a sum of n distinct primes).
For even n, a(n) <= A071148(n) (sum of the first n odd primes).

			The best solution for n=5 is (3,11,5,7,17) with a weight of 43. This is a prime vector because all the generated sums are prime: 3+11+5=19, 11+5+7=23, 5+7+17=29, 3+11+5+7+17=43.

Dmitry Kamenetsky, Prime sums of primes, arXiv:1703.06778 [math.HO], 2017.
Carlos Rivera, Puzzle 875: Vector of primes that generates distinct primes

Cf. A068873, A071148, A286269.

cp's OEIS Frontend

A102330 Triangle read by rows: n-th row consists of the lexicographically earliest set of n distinct primes whose sum is A068873(n).

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A007504 Sum of the first n primes.

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A073619 a(1) = 0; a(n) = smallest composite number which is a sum of n distinct primes.

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A383725 a(n) is the least number k such that omega(k) = n and the largest prime factor of k equals the sum of its remaining prime factors, where omega(k) = A001221(k).

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A286263 The smallest weight possible for a prime vector of order n.

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A073620 a(1) = 0; a(n) = smallest prime number which is a sum of n distinct composite numbers.

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A073621 Smallest composite number which is a sum of n distinct composite numbers.

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A100694 Smallest prime equal to the sum of exactly 2n+1 distinct odd primes.

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