A046992 -id:A046992 - cp's OEIS frontend

A073164 Quotients arising in A073162: A046992(n)/n if n is in A073162.

0, 1, 4, 7, 604, 7552, 28720, 43501, 176868, 3016559

Offset: 1

Labos Elemer, Jul 18 2002

			Sum of first 17 values of Pi(n) equals: 0+1+2+2+3+3+4+4+4+4+5+5+6+6+6+6+7 = 68 = 4*17, Sum(17)/17 = 4 = a(3).

Cf. A046992, A000720, A073162, A073163, A073224.

s = 0; Do[s = s + PrimePi[n]; If[ IntegerQ[s/n], Print[{n, s, s/n}]], {n, 1, 10^8}]

Integer values of s(n)=A046992(n)/n.

Edited and extended by Robert G. Wilson v, Jul 20 2002

a(10) from Donovan Johnson, Dec 15 2009

A122516 Primes in A046992.

Original entry on oeis.org

3, 5, 11, 19, 23, 37, 43, 61, 83, 107, 181, 271, 283, 349, 467, 499, 547, 563, 743, 821, 863, 947, 991, 1013, 1571, 2341, 2437, 2633, 2803, 2837, 2939, 3299, 3373, 3677, 3833, 4073, 4793, 4973, 5387, 5479, 5573, 6043, 6091, 6737, 7907, 8017, 8693, 8867

Offset: 1

Roger L. Bagula, Sep 16 2006

nonn

A subset of A057447. - Alexander Adamchuk, Sep 17 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A046992, A013918, A057447, A122933.

a122516 n = a122516_list !! (n-1)
a122516_list = filter ((== 1) . a010051) a046992_list
-- Reinhard Zumkeller, Feb 25 2012

Flatten[Table[If[PrimeQ[Sum[ PrimePi[n], {n, 1, m}]], Sum[PrimePi[n], {n, 1, m}], {}], {m, 1, 200}]]

a(n) = Prime[ A122933[n] ]. - Alexander Adamchuk, Sep 20 2006

Edited by N. J. A. Sloane, Sep 17 2006
More terms from Alexander Adamchuk, Sep 17 2006
Definition corrected, Sep 30 2006

A122515 a(n) = A007504(n) - A046992(n).

Original entry on oeis.org

2, 4, 7, 12, 20, 30, 43, 58, 77, 102, 128, 160, 195, 232, 273, 320, 372, 426, 485, 548, 613, 684, 758, 838, 926, 1018, 1112, 1210, 1309, 1412, 1528, 1648, 1774, 1902, 2040, 2180, 2325, 2476, 2631, 2792, 2958, 3126, 3303, 3482, 3665, 3850, 4046, 4254, 4466

Offset: 1

Roger L. Bagula, Sep 16 2006

nonn

Cf. A007504, A046992.

Table[Sum[Prime[n], {n, 1, m}] - Sum[PrimePi[n], {n, 1, m}], {m, 1, 50}]

from sympy import prime, primerange
def A122515(n): return -(n+1)*len(p:=list(primerange(n+1)))+(sum(p)<<1)+sum(primerange(n+1,prime(n)+1)) # Chai Wah Wu, Jan 01 2024

Edited by N. J. A. Sloane, Sep 17 2006

A034387 Sum of primes <= n.

Original entry on oeis.org

0, 2, 5, 5, 10, 10, 17, 17, 17, 17, 28, 28, 41, 41, 41, 41, 58, 58, 77, 77, 77, 77, 100, 100, 100, 100, 100, 100, 129, 129, 160, 160, 160, 160, 160, 160, 197, 197, 197, 197, 238, 238, 281, 281, 281, 281, 328, 328, 328, 328, 328, 328, 381, 381, 381, 381, 381

Offset: 1

N. J. A. Sloane

Also sum of all prime factors in n!.
For large n, these numbers can be closely approximated by the number of primes < n^2. For example, the sum of primes < 10^10 = 2220822432581729238. The number of primes < (10^10)^2 or 10^20 = 2220819602560918840. This has a relative error of 0.0000012743... - Cino Hilliard, Jun 08 2008
Equals row sums of triangle A143537. - Gary W. Adamson, Aug 23 2008
Partial sums of A061397. - Reinhard Zumkeller, Mar 21 2014

T. D. Noe, Table of n, a(n) for n = 1..10000
Jens Askgaard, On the additive period length of the Sprague-Grundy function of certain Nim-like games, arXiv:1902.06299 [math.CO], 2019. See p. 25.
Cino Hilliard, Sum of primes
Matt Visser, On the arithmetic average of the first n primes, arXiv:2505.04951 [math.NT], 2025. See p. 3.

Cf. A007504, A158662, A073837, A066779, A034386, A000720.

This is a lower bound on A287881.

a034387 n = a034387_list !! (n-1)
a034387_list = scanl1 (+) a061397_list
-- Reinhard Zumkeller, Mar 21 2014

a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+`if`(isprime(n), n, 0))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 29 2022

s=0; Table[s=s+n*Boole[PrimeQ[n]],{n,100}] (* Zak Seidov, Apr 11 2011 *)
Accumulate[Table[If[PrimeQ[n],n,0],{n,60}]] (* Harvey P. Dale, Jul 25 2016 *)

a(n)=sum(i=1,primepi(n),prime(i)) \\ Michael B. Porter, Sep 22 2009

a=0;for(k=1,100,print1(a=a+k*isprime(k),", ")) \\ Zak Seidov, Apr 11 2011

a(n) = if(n <= 1, return(0)); my(r=sqrtint(n)); my(V=vector(r, k, n\k)); my(L=n\r-1); V=concat(V, vector(L, k, L-k+1)); my(T=vector(#V, k, V[k]*(V[k]+1)\2)); my(S=Map(matrix(#V,2,x,y,if(y==1,V[x],T[x])))); forprime(p=2, r, my(sp=mapget(S,p-1), p2=p*p); for(k=1, #V, if(V[k] < p2, break); mapput(S, V[k], mapget(S,V[k]) - p*(mapget(S,V[k]\p) - sp)))); mapget(S,n)-1; \\ Daniel Suteu, Jun 29 2022

from sympy import isprime
from itertools import accumulate
def alist(n): return list(accumulate(k*isprime(k) for k in range(1, n+1)))
print(alist(57)) # Michael S. Branicky, Sep 18 2021

From the prime number theorem a(n) has the asymptotic expression: a(n) ~ n^2 / (2 log n). - Dan Fux (dan.fux(AT)OpenGaia.com), Apr 07 2001
a(n) = A158662(n) - 1. a(p) - a(p-1) = p, for p = primes (A000040), a(c) - a(c-1) = 0, for c = composite numbers (A002808). - Jaroslav Krizek, Mar 23 2009
a(n) = n^2/(2 log n) + O(n^2 log log n/log^2 n). - Vladimir Shevelev and Charles R Greathouse IV, May 29 2014
Conjecture: G.f.: Sum_{i>0} Sum_{j>=i} Sum_{k>=j|i-j+k is prime} x^k. - Benedict W. J. Irwin, Mar 31 2017
a(n) = (n+1)*A000720(n) - A046992(n). - Ridouane Oudra, Sep 18 2021
a(n) = A007504(A000720(n)). - Ridouane Oudra, Feb 22 2022
a(n) = Sum_{p<=n, p prime} p. - Wesley Ivan Hurt, Dec 31 2023

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

A034956 Divide natural numbers in groups with prime(n) elements and add together.

Original entry on oeis.org

3, 12, 40, 98, 253, 455, 850, 1292, 2047, 3335, 4495, 6623, 8938, 11180, 14335, 18815, 24249, 28731, 35845, 42884, 49348, 59408, 69139, 81791, 98164, 112211, 124939, 141026, 155434, 173681, 210439, 233966, 263040, 286062, 328098, 355152, 393442, 434558, 472777

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

Natural numbers starting from 1,2,3,4,...

			{1,2} #2 S=3;
{3,4,5} #3 S=12;
{6,7,8,9,10} #5 S=40;
{11,12,13,14,15,16,17} #7 S=98.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A006003, A027441, A034957.
Cf. A046992, A034958, A034959, A034960.
Cf. A000040, A000217, A007504.

s:= proc(n) s(n):= `if`(n<1, 0, s(n-1)+ithprime(n)) end:
a:= n-> (t-> t(s(n))-t(s(n-1)))(i-> i*(i+1)/2):
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 22 2023

Module[{nn=50,pr},pr=Prime[Range[nn]];Total/@TakeList[Range[ Total[ pr]], pr]](* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Oct 01 2017 *)

from itertools import islice
from sympy import nextprime
def A034956_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p+1)>>1
        a, p = a+p, nextprime(p)
A034956_list = list(islice(A034956_gen(),20)) # Chai Wah Wu, Mar 22 2023

From Hieronymus Fischer, Sep 27 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} k, n > 1.
a(n) = (A007504(n) - A007504(n-1))*(A007504(n) + A007504(n-1) + 1)/2, n > 1.
a(n) = (A000217(A007504(n)) - A000217(A007504(n-1))), n > 0.
If we define A007504(0) := 0, then the formulas above are also true for n=1.
a(n) = (A034960(n) + A000040(n))/2.
a(n) = A034957(n) + A000040(n). (End)

A034958 Divide primes into groups with prime(n) elements and add together.

Original entry on oeis.org

5, 23, 101, 311, 931, 1895, 3875, 6349, 10643, 18335, 25873, 39593, 55607, 71301, 94559, 127315, 167495, 204063, 258283, 315087, 369749, 451635, 533015, 640097, 779283, 902789, 1013795, 1159073, 1295871, 1457935, 1786691, 2002645, 2272221

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

			a(1) = 5 because the first 2 primes are 2 and 3 and 2 + 3 = 5.
a(2) = 23 because the next 3 primes are 5, 7, 11, and they add up to 23.
a(3) = 101 because the next 5 primes are 13, 17, 19, 23, 29 which add up to 101.
a(4) = 311 because the next 7 primes are 31, 37, 41, 43, 47, 53, 59 and they add up to 311.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A006003, A027441, A034956.

Cf. A007504, A046992, A034959, A034960, A180302.

Join[{5},Total[Prime[Range[#[[1]]+1,#[[2]]]]]&/@Partition[ Accumulate[ Prime[ Range[40]]],2,1]] (* Harvey P. Dale, Oct 03 2013 *)
Module[{nn=33},Total/@TakeList[Prime[Range[Total[Prime[Range[nn]]]]], Prime[ Range[ nn]]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 16 2018 *)
s = 0; Total[Table[s = s + 1; Prime[s], {j, 33}, {n, Prime[j]}], {2}] (* Horst H. Manninger, Jan 17 2019 *)

s(n) = sum(k=1, n, prime(k)); \\ A007504
a(n) = s(s(n)) - s(s(n-1)); \\ Michel Marcus, Oct 12 2018

From Hieronymus Fischer, Sep 26 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} A000040(k), n > 1.
a(n) = A007504(A007504(n)) - A007504(A007504(n-1)), n > 1.
If we define A007504(0) := 0, then the formulas are also true for n = 1.
(End)

A034960 Divide odd numbers into groups with prime(n) elements and add together.

Original entry on oeis.org

4, 21, 75, 189, 495, 897, 1683, 2565, 4071, 6641, 8959, 13209, 17835, 22317, 28623, 37577, 48439, 57401, 71623, 85697, 98623, 118737, 138195, 163493, 196231, 224321, 249775, 281945, 310759, 347249, 420751, 467801, 525943, 571985, 656047

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

			{1,3} #2 S=4;
{5,7,9} #3 S=21;
{11,13,15,17,19} #5 S=75;
{21,23,25,27,29,31,33} #7 S=189.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A006003, A027441, A034959.
Cf. A007504.
Cf. A000040, A034956, A034957, A034958, A046992, A061802.

S:= n-> sum(ithprime(k), k=1..n): seq(S(n+1)^2-S(n)^2, n=0..40); # Gary Detlefs, Dec 20 2011

Accumulate[Join[{2}, ListConvolve[{1, 1}, #]]]*# & [Prime[Range[50]]] (* Paolo Xausa, Jun 23 2025 *)

a0(n) = vecsum(primes(n))^2 - vecsum(primes(n-1))^2; \\ Michel Marcus, Jun 16 2024

from itertools import islice
from sympy import nextprime
def A034960_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p)
        a, p = a+p, nextprime(p)
A034960_list = list(islice(A034960_gen(),20)) # Chai Wah Wu, Mar 22 2023

From Hieronymus Fischer, Sep 26 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} (2*k-1).
a(n) = A007504(n)^2 - A007504(n-1)^2.
a(n) = 2*A034957(n) + A000040(n).
a(n) = 2*A034956(n) - A000040(n).
a(n) = A034959(n) + A000040(n). (End)
a(n) = A061802(n)*A000040(n). - Marco Zárate, May 12 2023

A034959 Divide even numbers into groups with prime(n) elements and add together.

Original entry on oeis.org

2, 18, 70, 182, 484, 884, 1666, 2546, 4048, 6612, 8928, 13172, 17794, 22274, 28576, 37524, 48380, 57340, 71556, 85626, 98550, 118658, 138112, 163404, 196134, 224220, 249672, 281838, 310650, 347136, 420624, 467670, 525806, 571846, 655898

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

			{0,2} #2 S=2;
{4,6,8} #3 S=18;
{10,12,14,16,18} #5 S=70;
{20,22,24,26,28,30,32} #7 S=182.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A006003, A027441, A034960.

Cf. A046992, A034956-A034958.

from itertools import islice
from sympy import nextprime
def A034959_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p-1)
        a, p = a+p, nextprime(p)
A034959_list = list(islice(A034959_gen(),20)) # Chai Wah Wu, Mar 22 2023

From Hieronymus Fischer, Sep 27 2012: (Start)
a(n) = 2*Sum_{k=(A007504(n-1)+1)..A007504(n)} (k-1), n > 1.
a(n) = (A007504(n) - A007504(n-1))*(A007504(n) + A007504(n-1) - 1), n > 1.
a(n) = 2*(A000217(A007504(n) - 1) - A000217(A007504(n-1) - 1)), n > 1.
If we define A007504(0):=0, then the formulas above are also true for n=1.
a(n) = 2*A034957(n).
a(n) = A034960(n) - A000040(n).
(End)

A368610 a(n) = Sum_{k=1..n} pi(k) * ceiling(n/k).

Original entry on oeis.org

0, 1, 4, 8, 14, 20, 30, 38, 49, 59, 72, 82, 101, 113, 130, 147, 167, 181, 206, 222, 248, 270, 293, 311, 346, 367, 392, 416, 448, 468, 508, 530, 565, 594, 624, 653, 700, 724, 757, 789, 836, 862, 913, 941, 985, 1028, 1067, 1097, 1159, 1193, 1240, 1279, 1328, 1360, 1418

Offset: 1

Wesley Ivan Hurt, Dec 31 2023

Cf. A000720 (pi), A046992, A092494, A368611, A368612, A368613.

Table[Sum[PrimePi[k] Ceiling[n/k], {k, n}], {n, 100}]

a(n) = A092494(n) + A368612(n).

cp's OEIS Frontend

A073164 Quotients arising in A073162: A046992(n)/n if n is in A073162.

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A122516 Primes in A046992.

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A122515 a(n) = A007504(n) - A046992(n).

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A034387 Sum of primes <= n.

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

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A034956 Divide natural numbers in groups with prime(n) elements and add together.

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A034958 Divide primes into groups with prime(n) elements and add together.

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