A111287 -id:A111287 - 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

A103208 Numbers k such that 3 divides prime(1) + ... + prime(k).

Original entry on oeis.org

10, 16, 18, 20, 24, 26, 28, 30, 32, 34, 36, 40, 42, 44, 46, 52, 54, 57, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 97, 99, 103, 105, 107, 111, 113, 119, 121, 123, 125, 127, 129, 134, 136, 138, 161, 163, 166, 169, 175, 177, 179, 185, 187, 195, 197, 199, 203, 205, 207, 211, 213

Offset: 1

Robert G. Wilson v, Mar 19 2005

nonn

Also, numbers k such that 3 divides the concatenation of the first k primes (see A019518).

The first comment and the description are true whenever the number of primes congruent to 1 mod 6 exceeds the number of primes congruent to 5 mod 6 and the difference is congruent to 1 mod 3 or the number of primes congruent to 5 mod 6 exceeds the number of primes congruent to 1 mod 6 and the difference is congruent to 2 mod 3. - Roderick MacPhee, Oct 30 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Hisanori Mishima, Smarandache consecutive prime sequences (n = 1 to 100).

Cf. A007504, A019518, A104644, A111287.

Cf. A111318, A111319, A111320, A111321, A111322, A111323, A111324, A111325, A111326, A111327.

s1:=[2]; M:=1000; for n from 2 to M do s1:=[op(s1),s1[n-1]+ithprime(n)]; od: s1;
f:=proc(k) global M,s1; local t1,n; t1:=[]; for n from 1 to M do if s1[n] mod k = 0 then t1:=[op(t1),n]; fi; od: t1; end; f(3);

f[n_] := FromDigits[ Flatten[ Table[ IntegerDigits[ Prime[i]], {i, n}]]]; Select[ Range[ 206], Mod[f[ # ], 3] == 0 &]
Flatten[Position[Accumulate[Prime[Range[250]]],?(Divisible[#,3]&)]] (* _Harvey P. Dale, Jan 14 2016 *)

a=0;b=0;for(x=3,1000,if(prime(x)%6==1,a+=1,b+=1);if((a-b)%3==1 || (b-a)%3==2,print1(x","))) \\ Roderick MacPhee, Oct 30 2015

lista(nn) = { s=0; for(k=1, nn, s += prime(k); if(s % 3 == 0, print1(k, ", ")););} \\ Altug Alkan, Dec 04 2015

Entry revised by N. J. A. Sloane, Nov 09 2005

A053050 a(n) = smallest integer m such that Sum_{k=1..m} prime(k) is divisible by n.

Original entry on oeis.org

1, 1, 10, 5, 2, 57, 5, 11, 20, 3, 8, 97, 49, 5, 57, 11, 4, 113, 23, 9, 40, 17, 23, 99, 9, 49, 26, 5, 7, 57, 39, 11, 76, 13, 180, 119, 29, 23, 119, 11, 6, 305, 10, 17, 242, 23, 39, 119, 40, 9, 179, 49, 25, 187, 17, 115, 70, 7, 30, 103, 151, 39, 40, 171, 131, 175, 38, 37, 52, 209, 19

Offset: 1

Felice Russo, Feb 25 2000

It follows from a theorem of Daniel Shiu that m always exists. See A111287 for details. - N. J. A. Sloane, Nov 05 2005

Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Cf. A007504, A111287, A002034, A011772.

Cf. A045985, A308749.

a053050 n = head [k | (k, x) <- zip [1..] a007504_list, mod x n == 0]
-- Reinhard Zumkeller, Oct 04 2015, Feb 10 2012

read transforms; M:=1000; p0:=[seq(ithprime(i),i=1..M)]; q0:=PSUM(p0); w:=[]; for n from 1 to M do p:=n; hit := 0; for i from 1 to M do if q0[i] mod p = 0 then w:=[op(w),i]; hit:=1; break; fi; od: if hit = 0 then break; fi; od: w;

Transpose[With[{aprs=Thread[{Range[500],Accumulate[Prime[Range[ 500]]]}]}, Flatten[Table[ Select[ aprs,Divisible[Last[#],n]&,1],{n,80}],1]]][[1]] (* Harvey P. Dale, Dec 14 2011 *)

A007504(a(n))/n = A308749(n). - Alois P. Heinz, Jun 21 2019

More terms from N. J. A. Sloane, Nov 05 2005

A111272 Where n-th prime occurs in A111267.

Original entry on oeis.org

3, 10, 2, 8, 17, 49, 4, 23, 41, 7, 39, 29, 6, 50, 79, 121, 30, 151, 38, 19, 267, 48, 174, 21, 287, 422, 240, 24, 94, 22, 16, 215, 861, 231, 143, 140, 396, 902, 18, 134, 340, 310, 269, 58, 12, 550, 479, 210, 229, 612, 221, 271, 194, 540, 145, 718, 168, 184, 90, 14, 272

Offset: 1

N. J. A. Sloane, Nov 03 2005

nonn

Cf. A111267, A111287.

A111267 := proc(nmin) local a,k,qn,dvs; a := [1] ; qn := 5 ; while nops(a) < nmin do dvs := sort([op(numtheory[divisors](qn))]) ; for k from 1 to nops(dvs) do if not dvs[k] in a then a := [op(a),dvs[k]] ; qn := qn+ithprime(nops(a)+1) ; break ; fi ; od ; od; RETURN(a) ; end: A111272 := proc(nmin) local a,a111267,n,i; a := [] ; a111267 := A111267(nmin) ; n := 1; while member( ithprime(n),a111267,'i') do a := [op(a),i] ; n := n+1 ; od; RETURN(a) ; end: A111272(1000) ; # R. J. Mathar, Aug 20 2007

More terms from R. J. Mathar, Aug 20 2007

A167790 a(n) is the index k of k-th prime prime(k) in the smallest sum s(k)=2+3+...+prime(k)=t*prime(n) of first k primes where t is a true divisor and first occurrence of factor prime(n) (n=1,2,3,...)

Original entry on oeis.org

3, 10, 3, 5, 8, 49, 13, 23, 23, 7, 39, 29, 15, 10, 39, 25, 30, 151, 38, 19, 139, 27, 174, 21, 287, 422, 240, 24, 94, 22, 16, 173, 861, 231, 143, 140, 213, 902, 18, 134, 143, 310, 70, 58, 295, 550, 237, 210, 229, 57, 221, 271, 194, 540, 145, 718, 116, 184, 90, 71, 168

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 12 2009, Nov 13 2009

nonn

It is conjectured that the sequence is infinite

If t is not restricted to nontrivial divisors, the sequence becomes A111287. - R. J. Mathar, Nov 17 2009

			s(5)=2+3+5+7+11=28=2^2*7=4*prime(4) gives a(4)=5 as first occurrence of prime factor prime(4)=7;
s(8)=2+3+5+7+11+13+17+19=77=7*11=7*prime(5) gives a(5)=8 as first occurrence of prime factor prime(5)=11;
s(422)=2+3+5+...+2917=570145= 5 * 101 * 1129=5645*prime(26) gives a(26)=422 and demonstrates the numerical difficulties.

Richard E. Crandall and Carl Pomerance, Prime Numbers, Springer 2005
Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005
Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996

Cf. A007504 (sum of first n primes).

Cf. A167764.

a(n) = min[2+3+...+prime(k)/t], where the minimum is taken with respect to k, the denominator t > 1 is an integer divisor of numerator s(k)=2+3+...+prime(k).

Extended by R. J. Mathar, Nov 17 2009

A168678 Least prime p such that the prime(n)-1 consecutive primes starting at p are all congruent to 1 (mod prime(n)).

Original entry on oeis.org

3, 31, 22501, 9984437

Offset: 1

T. D. Noe, Dec 02 2009

By a theorem of Shiu, a(n) exists for all n.

D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Cf. A111287.

Table[p=Prime[n]; cnt=0; q=2; While[q=NextPrime[q]; If[Mod[q,p]==1, cnt++, cnt=0]; cnt

A168678(n) = {local(p,m,c,r);p=2;r=2;m=prime(n);c=0;while(cMichael B. Porter, Feb 02 2010

A123119 Number of digits in sum of first n primes (A007504).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Jonathan Vos Post, Sep 28 2006

Since A007504(n) has the asymptotic expression ~ n^2 * log(n) / 2, a(n) has the asymptotic expression n^2 * log(n) / 2 = floor(log_10(10* n^2 * log(n) / 2)) = floor(log_10(5* n^2 * log(n))) = floor(log_10(5) + log_10(n^2) + log_10(log(n))) = floor(0.698970004 + 2*log_10(n) + log_10(log(n))). What is the smallest n such that a(n) = 5, 6, 7, ...?

			a(3) = 2 because 2 + 3 + 5 = 10 has 2 digits in its decimal expansion.

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

Cf. A000040, A000041, A004216, A004218, A034386, A055642, A111287.

f[n_] := Floor[ Log[10, Sum[Prime@i, {i, n}]] + 1]; Array[f, 105] (* Robert G. Wilson v *)
f[n_] := IntegerLength[Total[Prime[Range[n]]]]; Array[f, 105] (* Jan Mangaldan, Jan 04 2017 *)
IntegerLength/@Accumulate[Prime[Range[110]]] (* Harvey P. Dale, Jan 26 2019 *)

a(n) = A055642(A007504(n)) = floor(log_10(10*A007504(n))) = A004216(A007504(n)) + 1 = A004218(A007504(n) + 1).

More terms from Robert G. Wilson v, Oct 05 2006

cp's OEIS Frontend

A007504 Sum of the first n primes.

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A103208 Numbers k such that 3 divides prime(1) + ... + prime(k).

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A053050 a(n) = smallest integer m such that Sum_{k=1..m} prime(k) is divisible by n.

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A111272 Where n-th prime occurs in A111267.

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A167790 a(n) is the index k of k-th prime prime(k) in the smallest sum s(k)=2+3+...+prime(k)=t*prime(n) of first k primes where t is a true divisor and first occurrence of factor prime(n) (n=1,2,3,...)

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A168678 Least prime p such that the prime(n)-1 consecutive primes starting at p are all congruent to 1 (mod prime(n)).

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A123119 Number of digits in sum of first n primes (A007504).

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