A034387 -id:A034387 - cp's OEIS frontend

A053767 Sum of first n composite numbers.

0, 4, 10, 18, 27, 37, 49, 63, 78, 94, 112, 132, 153, 175, 199, 224, 250, 277, 305, 335, 367, 400, 434, 469, 505, 543, 582, 622, 664, 708, 753, 799, 847, 896, 946, 997, 1049, 1103, 1158, 1214, 1271, 1329, 1389, 1451, 1514, 1578, 1643, 1709, 1777, 1846, 1916, 1988

Offset: 0

G. L. Honaker, Jr., Mar 29 2000

nonn

a(A196415(n)) = A036691(A196415(n)) / A141092(n). - Reinhard Zumkeller, Oct 03 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
C. Rivera, See also

Cf. A002808, A051349.

First differences of A023539.

a053767 n = a053767_list !! (n-1)
a053767_list = scanl1 (+) a002808_list -- Reinhard Zumkeller, Oct 03 2011

A053767 := proc(n)
        add(A002808(i),i=1..n) ;
end proc: # R. J. Mathar, Oct 03 2011
ListTools[PartialSums](remove(isprime,[$2..1000])); # Robert Israel, Jan 09 2015

lst={};s=0;Do[If[ !PrimeQ[n], s=s+n;AppendTo[lst, s]], {n, 2, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Complement[Range[2,100],Prime[Range[PrimePi[100]]]]] (* Harvey P. Dale, Dec 28 2010 *)
Accumulate[Select[Range[2, 100], ! PrimeQ[#] &]]

lista(nn) = {my(s=0); forcomposite(n=0, nn, print1(s, ", "); s += n;);} \\ Michel Marcus, Jan 09 2015

a(n) = A000217(A002808(n)) - A034387(A002808(n)) - 1 . - Robert Israel, Jan 09 2015

a(n) = A051349(n+1) - 1. - Michel Marcus, Feb 16 2018

a(0)=0 prepended by Max Alekseyev, Feb 10 2018

A046731 a(n) = sum of primes < 10^n.

Original entry on oeis.org

0, 17, 1060, 76127, 5736396, 454396537, 37550402023, 3203324994356, 279209790387276, 24739512092254535, 2220822432581729238, 201467077743744681014, 18435588552550705911377, 1699246443377779418889494, 157589260710736940541561021, 14692398516908006398225702366

Offset: 0

Enoch Haga

a(21) was already correctly computed by Marc Deleglise in 2009 but in 2011 he withdrew his result because his verification failed. - Kim Walisch, Jun 06 2016

			The primes less than 10 give 2+3+5+7 = 17.

Lorenzo Pieri, Table of n, a(n) for n = 0..26 [terms a(0)-a(20) from Marc Deleglise; terms a(21)-a(23) from Kim Walisch; terms a(24)-a(25) from David Baugh]
M. Deleglise and J. Rivat, Computing pi(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math. Comp., 65 (1996), 235-245.
Cino Hilliard, GmpDemo Sumprimes.
Cino Hilliard, Achim Sieve Gmp Sumprimes.
Cino Hilliard, Achim Multi-Prec add Sumprimes.
Kim Walisch, primesum program.

Cf. A034387.

Join[{0, s = 17}, Table[Do[If[PrimeQ[i], s += i], {i, 10^n + 1, 10^(n + 1), 2}]; s, {n, 7}]] (* Jayanta Basu, Jun 28 2013 *)
Table[Sum[Prime[i], {i, PrimePi[10^n]}], {n, 0, 7}]  (* Kim Walisch, Dec 21 2017 *)

a(n) = my(s=0); forprime(p=1, 10^n, s += p); s; \\ Michel Marcus, Jan 14 2015

use ntheory ":all"; say "$ ",sum_primes(10**$) for 0..15; # Dana Jacobsen, May 04 2017

a(n) is about 100^n/(n log 100). - Charles R Greathouse IV, Jan 29 2013

a(n) = Sum_{i=2..10^n} A061397(i). - José de Jesús Camacho Medina, Aug 08 2016

Corrected and extended by Jud McCranie
a(12) and a(13) from Cino Hilliard, Aug 14 2006
New value for a(13) from Cino Hilliard, Oct 24 2007
There was indeed an error in a(13) both in the entry here and in the b-file. This has now been corrected. - N. J. A. Sloane, Nov 23 2007
Two new values from Marc Deleglise, May 21 2008 - see the b-file.
a(21) from Marc Deleglise, Jun 29 2008 - see the b-file.
Nov 15 2011: Marc Deleglise has withdrawn his value for a(21).
a(21)-a(22) from Kim Walisch, Jun 06 2016
a(23) from Kim Walisch, Jun 11 2016
a(24) from David Baugh using Kim Walisch's primesum program, Jun 17 2016
a(25) from David Baugh using Kim Walisch's primesum program, Oct 16 2016
a(26) from Kim Walisch, May 25 2022, added by Lorenzo Pieri

A024924 a(n) = Sum_{k=1..n} prime(k)*floor(n/prime(k)).

Original entry on oeis.org

0, 0, 2, 5, 7, 12, 17, 24, 26, 29, 36, 47, 52, 65, 74, 82, 84, 101, 106, 125, 132, 142, 155, 178, 183, 188, 203, 206, 215, 244, 254, 285, 287, 301, 320, 332, 337, 374, 395, 411, 418, 459, 471, 514, 527, 535, 560, 607, 612, 619, 626, 646, 661, 714, 719, 735, 744, 766, 797, 856

Offset: 0

Clark Kimberling

nonn

For n > 2, sum of all distinct prime factors composing numbers from 2 to n.

M. Kalecki, On certain sums extended over primes or prime factors (in Polish), Prace Mat., Vol. 8 (1963/64), pp. 121-129.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 144.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Plot of a(n) / ((Pi^2/12) * n^2/log(n)) for n = 1..10^8

Partial sums of A008472.

Cf. A000720, A024934, A034387.

[0] cat [ &+[ NthPrime(k)*Floor(n/NthPrime(k)): k in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Jul 28 2019

Join[{0}, Table[Sum[Prime[k] Floor[n / Prime[k]], {k, 1, n}], {n, 1, 60}]] (* Vincenzo Librandi, Jul 28 2019 *)
Join[{0}, Accumulate[Table[Sum[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}]]] (* Vaclav Kotesovec, May 20 2020 *)

a(n) = sum(k=1, n, prime(k)*(n\prime(k))); \\ Michel Marcus, Mar 01 2015

a(n)=my(s); forprime(p=2,n, s+=n\p*p); s \\ Charles R Greathouse IV, Jun 26 2020

from sympy import prime
def A024924(n): return sum((p:=prime(k))*(n//p) for k in range(1,n+1)) # Chai Wah Wu, Sep 18 2023

a(n) = n*A000720(n) - A024934(n). - Max Alekseyev, Feb 10 2012
a(n) = A034387([n/1]) + A034387([n/2]) + ... + A034387([n/n]). Terms can be computed efficiently with the following formula: a(n) = A034387([n/1]) + ... + A034387([n/m]) - m*A034387([n/m]) + Sum_{prime p<=n/m} p*[n/p], where m = [sqrt(n)]. - Max Alekseyev, Feb 10 2012
G.f.: Sum_{k >=1} (prime(k)*x^prime(k)/(1-x^prime(k)))/(1-x). - Vladeta Jovovic, Aug 11 2004
a(n) ~ ((Pi^2 + o(1))/12) * n^2/log(n) (Kalecki, 1963/64). - Amiram Eldar, Mar 04 2021

a(0)=0 prepended by Max Alekseyev, Feb 10 2012

A073837 Sum of primes p satisfying n <= p <= 2n.

Original entry on oeis.org

2, 5, 8, 12, 12, 18, 31, 24, 41, 60, 60, 72, 72, 59, 88, 119, 119, 102, 139, 120, 161, 204, 204, 228, 228, 228, 281, 281, 281, 311, 372, 341, 341, 408, 408, 479, 552, 515, 515, 594, 594, 636, 636, 593, 682, 682, 682, 635, 732, 732, 833, 936, 936, 990, 1099, 1099

Offset: 1

Amarnath Murthy, Aug 12 2002

nonn

a(n) = A034387(2*n) - A034387(n-1); a(n) <= A179213(n). [Reinhard Zumkeller, Jul 05 2010]

			a(7) = 31 = 7+11+13 (sum of primes between 7 and 14).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A073838.

for n from 1 to 150 do l := 0:for j from n to 2*n do if isprime(j) then l := l+j:fi:od:a[n] := l:od:seq(a[j],j=1..150);

Table[Total[Select[Range[n, 2 n], PrimeQ]], {n, 56}] (* Jayanta Basu, Aug 12 2013 *)

PARI
```
a(n)=sum(i=n,2*n,i*isprime(i))
```

More terms from Sascha Kurz and Benoit Cloitre, Aug 14 2002

A066779 Sum of squarefree numbers <= n.

Original entry on oeis.org

1, 3, 6, 6, 11, 17, 24, 24, 24, 34, 45, 45, 58, 72, 87, 87, 104, 104, 123, 123, 144, 166, 189, 189, 189, 215, 215, 215, 244, 274, 305, 305, 338, 372, 407, 407, 444, 482, 521, 521, 562, 604, 647, 647, 647, 693, 740, 740, 740, 740, 791, 791, 844, 844, 899, 899

Offset: 1

Benoit Cloitre, Jan 18 2002

D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A008966, A013928, A034387, A179213, A179215.

Table[ n*Boole[ SquareFreeQ[n] ], {n, 1, 56}] // Accumulate (* Jean-François Alcover, Jun 18 2013 *)

s=0; for (n=1, 1000, write("b066779.txt", n, " ", s+=moebius(n)^2*n) ) \\ Harry J. Smith, Mar 24 2010

a(n)=sum(d=1,sqrtint(n),moebius(d)*d^2*binomial(n\d^2+1,2)) \\ Charles R Greathouse IV, Apr 26 2012

a(n)=my(s,k2); forsquarefree(k=1,sqrtint(n), k2=k[1]^2; s+= k2*binomial(n\k2+1,2)*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

from sympy.ntheory.factor_  import core
def a(n): return sum ([i for i in range(1, n + 1) if core(i) == i]) # Indranil Ghosh, Apr 16 2017

a(n) = Sum_{i=1..n} mu(i)^2*i.
a(n) = Sum_{k=1..n} k*A008966(k). - Reinhard Zumkeller, Jul 05 2010
a(n) = Sum_{d=1..sqrt(n)} mu(d)*d^2*floor(n/d^2)*floor(n/d^2+1)/2. - Charles R Greathouse IV, Apr 26 2012
G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 - x). - Ilya Gutkovskiy, Apr 16 2017
a(n) ~ (3/Pi^2) * n^2 + O(n^(3/2)) (Suryanarayana, 1969). - Amiram Eldar, Mar 07 2021

A063956 Sum of unitary prime divisors of n.

Original entry on oeis.org

0, 2, 3, 0, 5, 5, 7, 0, 0, 7, 11, 3, 13, 9, 8, 0, 17, 2, 19, 5, 10, 13, 23, 3, 0, 15, 0, 7, 29, 10, 31, 0, 14, 19, 12, 0, 37, 21, 16, 5, 41, 12, 43, 11, 5, 25, 47, 3, 0, 2, 20, 13, 53, 2, 16, 7, 22, 31, 59, 8, 61, 33, 7, 0, 18, 16, 67, 17, 26, 14, 71, 0, 73, 39, 3, 19, 18, 18, 79, 5, 0

Offset: 1

Labos Elemer, Sep 04 2001

			The prime divisors of 420 = 2^2 * 3 * 5 * 7. Among them, those that have exponent 1 (i.e., unitary prime divisors) are {3, 5, 7}, so a(420) = 3 + 5 + 7 = 15.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A008472, A034387, A034444, A056169, A056170, A056171, A056172, A063958.

Table[DivisorSum[n, # &, And[PrimeQ@ #, GCD[#, n/#] == 1] &], {n, 81}] (* Michael De Vlieger, Feb 17 2019 *)
f[p_, e_] := If[e == 1, p, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)
Join[{0},Table[Total[Select[FactorInteger[n],#[[2]]==1&][[;;,1]]],{n,2,100}]] (* Harvey P. Dale, Jan 26 2025 *)

{ for (n=1, 1000, f=factor(n)~; a=0; for (i=1, length(f), if (f[2, i]==1, a+=f[1, i])); write("b063956.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009

a(n*m) = a(n) + a(m) - a(gcd(n^2, m)) - a(gcd(n, m^2)) for all n and m > 0 (conjecture). - Velin Yanev, Feb 17 2019
From Amiram Eldar, Jul 24 2024: (Start)
a(n) = A008472(n) - A063958(n).
Additive with a(p^e) = p is e = 1, and 0 otherwise. (End)

Example clarified by Harvey P. Dale, Jan 26 2025

A101203 a(n) = sum of nonprimes <= n.

Original entry on oeis.org

0, 1, 1, 1, 5, 5, 11, 11, 19, 28, 38, 38, 50, 50, 64, 79, 95, 95, 113, 113, 133, 154, 176, 176, 200, 225, 251, 278, 306, 306, 336, 336, 368, 401, 435, 470, 506, 506, 544, 583, 623, 623, 665, 665, 709, 754, 800, 800, 848, 897, 947, 998, 1050, 1050, 1104, 1159, 1215

Offset: 0

Reinhard Zumkeller, Jan 23 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A062298, A018252.
Partial sums of A191558.
Cf. A000217, A034387, A101256.

a101203 n = a101203_list !! (n-1)
a101203_list = scanl (+) 0 $ zipWith (*) [1..] $ map (1 -) a010051_list
-- Reinhard Zumkeller, Oct 10 2013

Accumulate[Table[If[PrimeQ[n],0,n],{n,0,60}]] (* Harvey P. Dale, Oct 02 2020 *)

my(s=0); for(k=0,100,if(!isprime(k),s+=k);print1(s", ")); \\ Cino Hilliard, Feb 04 2006

a(n) = A000217(n) - A034387(n) = A101256(n) + 1.

A158662 Sum of primes <= n if 1 is counted as a prime.

Original entry on oeis.org

1, 3, 6, 6, 11, 11, 18, 18, 18, 18, 29, 29, 42, 42, 42, 42, 59, 59, 78, 78, 78, 78, 101, 101, 101, 101, 101, 101, 130, 130, 161, 161, 161, 161, 161, 161, 198, 198, 198, 198, 239, 239, 282, 282, 282, 282, 329, 329, 329, 329, 329, 329

Offset: 1

Jaroslav Krizek, Mar 23 2009, Apr 12 2009

nonn

a(n) = A034387(n) + 1. a(p) - a(p-1) = p, for p = primes (A000040), a(c) - a(c-1) = 0, for c = composite numbers (A002808).

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

Cf. A034387, A000040, A002808.

Module[{nn=60,c},c=Join[{1},Prime[Range[PrimePi[nn]]]];Table[Total[ Select[ c,#<=n&]],{n,nn}]] (* Harvey P. Dale, Jun 01 2014 *)
Accumulate[Join[{1},Table[If[PrimeQ[n],n,0],{n,60}]]] (* Harvey P. Dale, Feb 27 2017 *)

A182936 Greatest common divisor of the proper divisors of n, 0 if there are none.

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 0, 2, 3, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 1, 5, 1, 3, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 7, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 3, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Peter Luschny, Mar 22 2011

nonn

Here a proper divisor d of n is a divisor of n such that 1 < d < n.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A014963, A020500, A048671, A034387, A061397, A072107, A380118.

A182936 := n -> igcd(op(numtheory[divisors](n) minus {1,n}));
seq(A182936(i), i=1..79); # Peter Luschny, Mar 22 2011

Join[{0}, Table[GCD@@Most[Rest[Divisors[n]]],{n,2,110}]] (* Harvey P. Dale, May 04 2018 *)
(* From Peter Luschny, Jan 31 2025: (Start) *)
Join[{0}, Table[Exp[MangoldtLambda[n]] - If[PrimeQ[n], n, 0], {n,2,110}]]
(* or *)
Table[Cyclotomic[n, 1] - If[PrimeQ[n], n, 0], {n,1,110}] (* End *)

A182936(n) = { my(divs=divisors(n)); if(#divs<3,0,gcd(vector(numdiv(n)-2,k,divs[k+1]))); }; \\ Antti Karttunen, Sep 23 2017

a(n) = 0 if n is not composite, p if n is a proper power of prime p, and 1 otherwise. - Franklin T. Adams-Watters, Mar 22 2011
Conjecture: Sum_{k=1..n} a(k) = A072107(n) - A034387(n) - 1. - Vaclav Kotesovec, Jan 29 2025
From Peter Luschny, Jan 31 2025: (Start)
a(n) = A014963(n) - A061397(n) for n > 1. In other words, this sequence is the exponential von Mangoldt function restricted to proper divisors of n. See A380118. This implies the above conjecture.
a(n) = A020500(n) - A061397(n). (End)

More terms from Antti Karttunen, Sep 23 2017

A051352 a(0) = 0; for n>0, a(n) = a(n-1) + n if n not prime else a(n-1) - n.

Original entry on oeis.org

0, 1, -1, -4, 0, -5, 1, -6, 2, 11, 21, 10, 22, 9, 23, 38, 54, 37, 55, 36, 56, 77, 99, 76, 100, 125, 151, 178, 206, 177, 207, 176, 208, 241, 275, 310, 346, 309, 347, 386, 426, 385, 427, 384, 428, 473, 519, 472, 520, 569, 619, 670, 722, 669, 723, 778

Offset: 0

Armand Turpel armandt(AT)unforgettable.com

Sequence is not monotonic.

Difference between sum of nonprime numbers and prime numbers <= n. - Zak Seidov, Sep 27 2003

William A. Tedeschi, Table of n, a(n) for n = 0..10000

Cf. A005171, A010051.

Cf. A000217, A034387.

a051352 n = a051352_list !! n
a051352_list = 0 : zipWith (+)
   (a051352_list) (zipWith (*) [1..] $ map ((1 -) . (* 2)) a010051_list)
-- Reinhard Zumkeller, Jan 02 2015

A034387 := proc(n)
    option remember;
    if n <= 1 then
        0;
    else
        procname(n-1)+ `if`(isprime(n), n, 0)
    end if;
end proc:
A051352 := proc(n)
    n*(n+1)/2 - 2*A034387(n) ;
end proc:
seq(A051352(n),n=0..40) ; # R. J. Mathar, Jun 26 2024

a[0]=0;a[n_]:=a[n]=If[PrimeQ[n],a[n-1]-n,a[n-1]+n]; Table[a[i], {i,0,60}] (* Harvey P. Dale, Apr 07 2011 *)
nxt[{n_,a_}]:={n+1,If[PrimeQ[n+1],a-n-1,a+n+1]}; NestList[nxt,{0,0},60][[All,2]] (* Harvey P. Dale, Sep 07 2022 *)

a(n) = my(v=primes([1, n])); n*(n+1)/2 -2*vecsum(v); \\ Michel Marcus, Jun 24 2024

a(n) = a(n-1) + n * (1 - 2*A010051(n)) = a(n-1) + n * (2*A005171(n) - 1) = a(n-1) + n * (A005171(n) - A010051(n)). - Reinhard Zumkeller, Nov 25 2009

a(n) = A000217(n) - 2*A034387(n). - Michel Marcus, Jun 24 2024

cp's OEIS Frontend

A053767 Sum of first n composite numbers.

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A046731 a(n) = sum of primes < 10^n.

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A024924 a(n) = Sum_{k=1..n} prime(k)*floor(n/prime(k)).

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A073837 Sum of primes p satisfying n <= p <= 2n.

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A066779 Sum of squarefree numbers <= n.

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A063956 Sum of unitary prime divisors of n.

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