A033286 -id:A033286 - 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

A014689 a(n) = prime(n)-n, the number of nonprimes less than prime(n).

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 10, 11, 14, 19, 20, 25, 28, 29, 32, 37, 42, 43, 48, 51, 52, 57, 60, 65, 72, 75, 76, 79, 80, 83, 96, 99, 104, 105, 114, 115, 120, 125, 128, 133, 138, 139, 148, 149, 152, 153, 164, 175, 178, 179, 182, 187, 188, 197, 202, 207, 212, 213, 218, 221, 222

Offset: 1

Mohammad K. Azarian

a(n) = A048864(A000040(n)) = number of nonprimes in RRS of n-th prime. - Labos Elemer, Oct 10 2002
A000040 - A014689 = A000027; in other words, the sequence of natural numbers subtracted from the prime sequence produces A014689. - Enoch Haga, May 25 2009
a(n) = A000040(n) - n. a(n) = inverse (frequency distribution) sequence of A073425(n), i.e., number of terms of sequence A073425(n) less than n. a(n) = A065890(n) + 1, for n >= 1. a(n) - 1 = A065890(n) = the number of composite numbers, i.e., (A002808) less than n-th primes, (i.e., < A000040(n)). - Jaroslav Krizek, Jun 27 2009
a(n) = A162177(n+1) + 1, for n >= 1. a(n) - 1 = A162177(n+1) = the number of composite numbers, i.e., (A002808) less than (n+1)-th number of set {1, primes}, (i.e., < A008578(n+1)). - Jaroslav Krizek, Jun 28 2009
Conjecture: Each residue class contains infinitely many terms of this sequence. Similarly, for any integers m > 0 and r, we have prime(n) + n == r (mod m) for infinitely many positive integers n. - Zhi-Wei Sun, Nov 25 2013
First differences are A046933 = differences minus one between successive primes. - Gus Wiseman, Jan 18 2020

T. D. Noe, Table of n, a(n) for n=1..1000

Equals A014692 - 1.
Cf. A000040, A033286, A158611, A002808, A065890.
Cf. A232463, A232443.
The sum of prime factors of n is A001414(n).
The sum of prime indices of n is A056239(n).
Their difference is A331415(n).
Cf. A000720, A046933, A056239, A318995, A325036, A331380, A331416, A331418.

a014689 n = a000040 n - fromIntegral n
-- Reinhard Zumkeller, Apr 09 2012

[NthPrime(n)-n: n in [1..70]]; // Vincenzo Librandi, Mar 20 2013

Table[Prime[n] - n, {n, 61}] (* Alonso del Arte *)

a(n) = prime(n)-n \\ Charles R Greathouse IV, Sep 05 2011

from sympy import prime
def A014689(n): return prime(n)-n # Chai Wah Wu, Oct 11 2024

G.f: b(x) - x/((1-x)^2), where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 13 2016

More terms from Vasiliy Danilov (danilovv(AT)usa.net), Jul 1998

Correction for Aug 2009 change of offset in A158611 and A008578 by Jaroslav Krizek, Jan 27 2010

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

A152535 a(n) = n*prime(n) - Sum_{i=1..n} prime(i).

Original entry on oeis.org

0, 1, 5, 11, 27, 37, 61, 75, 107, 161, 181, 247, 295, 321, 377, 467, 563, 597, 705, 781, 821, 947, 1035, 1173, 1365, 1465, 1517, 1625, 1681, 1797, 2217, 2341, 2533, 2599, 2939, 3009, 3225, 3447, 3599, 3833, 4073, 4155, 4575, 4661

Offset: 1

Omar E. Pol, Dec 06 2008

a(n) is also the area under the curve of the function pi(x) from 0 to prime(n). - Omar E. Pol, Nov 13 2013

			From _Omar E. Pol_, Apr 27 2015: (Start)
For n = 5 the 5th prime is 11 and the sum of first five primes is 2 + 3 + 5 + 7 + 11 = 28, so a(5) = 5*11 - 28 = 27.
Illustration of a(5) = 27:
Consider a diagram in the first quadrant of the square grid in which the number of cells in the n-th horizontal bar is equal to the n-th prime, as shown below:
.      _ _ _ _ _ _ _ _ _ _ _
. 11  |_ _ _ _ _ _ _ _ _ _ _|
.  7  |_ _ _ _ _ _ _|* * * *
.  5  |_ _ _ _ _|* * * * * *
.  3  |_ _ _|* * * * * * * *
.  2  |_ _|* * * * * * * * *
.
a(5) is also the area (or the number of cells, or the number of *'s) under the bar's structure of prime numbers: a(5) = 1 + 4 + 6 + 16 = 27.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Christian Axler, On a sequence involving the prime numbers, arXiv:1504.04467 [math.NT], 2015 and J. Int. Seq. 18 (2015) # 15.7.6.
Christian Axler, Improving the Estimates for a Sequence Involving Prime Numbers, arXiv:1706.04049 [math.NT], 2017.

Cf. A000040, A000720, A001223, A006093, A007504, A033286, A046992, A090942, A117495, A141042 (first differences), A189892, A230849, A230850, A247380.

nn = 100; p = Prime[Range[nn]]; Range[nn] p - Accumulate[p] (* T. D. Noe, May 02 2011 *)

vector(80, n, n*prime(n) - sum(k=1, n, prime(k))) \\ Michel Marcus, Apr 20 2015

from sympy import prime, primerange
def A152535(n): return (n-1)*(p:=prime(n))-sum(primerange(p)) # Chai Wah Wu, Jan 01 2024

[n*nth_prime(n) - sum(nth_prime(j) for j in range(1,n+1)) for n in range(1,45)] # Danny Rorabaugh, Apr 18 2015

a(n) = A033286(n) - A007504(n). - Omar E. Pol, Aug 09 2012
a(n) = A046992(A006093(n)). - Omar E. Pol, Apr 21 2015
a(n+1) = Sum_{k=A000124(n-1)..A000217(n)} A204890(k). - Benedict W. J. Irwin, May 23 2016
a(n) = Sum_{k=1..n-1} k*A001223(k). - François Huppé, Mar 16 2022

A076146 a(1) = 1; a(n) = a(n-1)*prime(a(n-1)).

Original entry on oeis.org

1, 2, 6, 78, 30966, 11234495766, 3197149582479668022558

Offset: 1

Zak Seidov, Nov 02 2002

Previous term * prime(previous term). Previous term + prime(previous term) is A074271.

Matula-Goebel numbers of the finite ordinal numbers; see also A007097. - Gus Wiseman, Aug 30 2016

Cf. A033286, A074271, A007097, A000040, A004111, A348067.

NestList[# Prime@ # &, 1, 6] (* Michael De Vlieger, Aug 30 2016 *)

a(7) from Gus Wiseman, Aug 30 2016

A084122 Numbers k such that k*prime(k) is a palindrome.

Original entry on oeis.org

1, 2, 5, 12, 16, 3623, 4119, 618725, 708567, 1498739, 2762990591

Offset: 1

Giovanni Resta, May 14 2003

a(12) > 3.7*10^12. - Giovanni Resta, Jun 28 2013

			4119 is in the sequence since the 4119th prime is 39119 and 4199*39119 = 161131161 is a palindrome.

Cf. A002113, A033286, A084121, A084123.

ispal:= proc(n) local L;
  L:= convert(n,base,10);
  L = ListTools:-Reverse(L);
end proc:
R:= NULL: count:= 0: p:= 1:
for k from 1 while count < 11 do
  p:= nextprime(p);
  if ispal(k*p) then R:= R,k; count:= count+1 fi
od:
R; # Robert Israel, Feb 22 2023

palQ[n_]:=FromDigits[Reverse[IntegerDigits[n]]]==n; t={}; Do[If[palQ[Prime[n]*n],AppendTo[t,n]],{n,15*10^5}]; t (* Jayanta Basu, May 11 2013 *)

ispal(n) = my(d=digits(n)); d == Vecrev(d); \\ A002113
isok(k) = ispal(k*prime(k)) \\ Alexandru Petrescu, Feb 22 2023

from sympy import sieve
def ok(n): return n and (s := str(n*sieve[n])) ==  s[::-1]
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Feb 22 2023

A105455 Numbers k such that kprime(k)+(k+1)prime(k+1)+(k+2)*prime(k+2) is prime.

Original entry on oeis.org

1, 6, 12, 20, 22, 24, 28, 30, 34, 56, 60, 142, 144, 148, 168, 192, 196, 230, 252, 260, 276, 282, 304, 322, 334, 344, 346, 352, 366, 374, 380, 386, 394, 404, 418, 424, 432, 440, 444, 470, 478, 484, 572, 590, 610, 612, 630, 642, 662, 684, 754, 766, 784, 790, 840, 842, 874, 886

Offset: 1

Zak Seidov, May 02 2005

nonn

			k=1: 1*prime(1) + 2*prime(2) + 3*prime(3) = 1*2 + 2*3 + 3*5 = 23 prime,
k=6: 6*prime(6) + 7*prime(7) + 8*prime(8) = 6*13 + 7*17 + 8*19 = 349 prime. - _Zak Seidov_, Feb 18 2016

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A033286, A105454, A152117, A119487.

[n: n in [1..1000] | IsPrime(n*NthPrime(n)+(n+1)*NthPrime(n+1)+(n+2)*NthPrime(n+2))]; // Vincenzo Librandi, Feb 06 2016

bb={};Do[If[PrimeQ[n Prime[n]+(n+1) Prime[n+1]+(n+2) Prime[n+2]], bb=Append[bb, n]], {n, 1, 400}];bb
Select[Range@ 900, PrimeQ[# Prime[#] + (# + 1) Prime[# + 1] + (# + 2) Prime[# + 2]] &] (* Michael De Vlieger, Feb 05 2016 *)

lista(nn) = {for(n=1, nn, if(ispseudoprime(n*prime(n)+(n+1)*prime(n+1)+(n+2)*prime(n+2)), print1(n, ", "))); } \\ Altug Alkan, Feb 05 2016

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    m, p, q, r = 1, 2, 3, 5
    while True:
        t = m*p + (m+1)*q + (m+2)*r
        if isprime(t): yield m
        m, p, q, r = m+1, q, r, nextprime(r)
print(list(islice(agen(), 58))) # Michael S. Branicky, May 17 2022

A152117 a(n) = n(n-th prime) + (n+1)((n+1)-th prime).

Original entry on oeis.org

8, 21, 43, 83, 133, 197, 271, 359, 497, 631, 785, 977, 1135, 1307, 1553, 1851, 2101, 2371, 2693, 2953, 3271, 3647, 4045, 4561, 5051, 5407, 5777, 6157, 6551, 7327, 8129, 8713, 9247, 9941, 10651, 11245, 12003, 12707, 13433, 14259, 14941, 15815, 16705

Offset: 1

Klaus Brockhaus, Dec 10 2008

nonn

a(n) = A033286(n) + A033286(n+1).

			5*(fifth prime) + 6*(sixth prime) = 5*11 + 6*13 = 55 + 78 = 133.

Klaus Brockhaus, Table of n, a(n) for n=1..1000

Cf. A000040 (prime numbers), A033286 (n*(n-th prime)), A033287 (first differences of A033286), A119487 (primes in this sequence).

[ n*NthPrime(n)+(n+1)*NthPrime(n+1): n in [1..43] ];

Total/@Partition[Times@@@Table[{n,Prime[n]},{n,50}],2,1] (* Harvey P. Dale, Aug 13 2019 *)

a(n) = n*prime(n) + (n+1)*prime(n+1); \\ Michel Marcus, Feb 05 2016

The convergence is very slow, need to use the first 100000000 primes to obtain the correct value of the coefficient of 10^(-23).

The constant is in the interval [0.28417070547086825017714, 0.28417070547086825017743]; these safe limits are computed by accumulating in parallel the partial sum of the lower estimate 1/n^4 = Zeta(4). - R. J. Mathar, Feb 06 2015

cp's OEIS Frontend

A033287 First differences of A033286.

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A253634 Decimal expansion of Sum_{n>=1} 1/A033286(n)^2.

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A007504 Sum of the first n primes.

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A014689 a(n) = prime(n)-n, the number of nonprimes less than prime(n).

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

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A152535 a(n) = n*prime(n) - Sum_{i=1..n} prime(i).

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A076146 a(1) = 1; a(n) = a(n-1)*prime(a(n-1)).

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A105455 Numbers k such that kprime(k)+(k+1)prime(k+1)+(k+2)*prime(k+2) is prime.

A152117 a(n) = n(n-th prime) + (n+1)((n+1)-th prime).