A101256 -id:A101256 - cp's OEIS frontend

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

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.

A364879 a(n) is the smallest number k such that (sum of composites <= k) / (sum of primes <= k) >= n.

Original entry on oeis.org

2, 6, 10, 28, 126, 520, 1394, 4440, 11765, 35702, 98202, 271718, 736814, 2012631, 5478367, 14867499, 40448112, 109944053, 298170203, 810416222, 2200884471, 5980529528

Offset: 0

Jon E. Schoenfield, Sep 10 2023

a(n)+1 is a prime for n = 0, 1, 2, 3, 4, 5, and 7 (thus, for n = 1, 2, 3, 4, 5, and 7, a(n) is the last of a run of consecutive composites), but not for n = 6, nor for any n in 8..16.

For n > 0, a(n) is at least the n-th in a run of consecutive composites. a(15) is the 58th in a run of 71 consecutive composites.

			Let Sp(k) and Sc(k) be the sums of the primes <= k and the composites <= k, respectively. Then the sums and ratios begin as follows:
.
   k | Sp(k) | Sc(k) | Sc(k)/Sp(k)
  ---+-------+-------+------------
   1 |     0 |     0 | (undefined)
   2 |     2 |     0 |  0/2  = 0         so a(0) =  2
   3 |     5 |     0 |  0/5  = 0
   4 |     5 |     4 |  4/5  = 0.8
   5 |    10 |     4 |  4/10 = 0.4
   6 |    10 |    10 | 10/10 = 1         so a(1) =  6
   7 |    17 |    10 | 10/17 = 0.5882...
   8 |    17 |    18 | 18/17 = 1.0588...
   9 |    17 |    27 | 27/17 = 1.5882...
  10 |    17 |    37 | 37/17 = 2.1764... so a(2) = 10

Cf. A000040, A002808, A007504, A034387, A053767, A101256.

from itertools import count
from sympy import isprime
def A364879(n):
    c, cn, m = 0, 0, n+1<<1
    for k in count(2):
        if isprime(k):
            c += k
            cn += k*m
        if k*(k+1)-1 >= cn:
            return k # Chai Wah Wu, Sep 10 2023

a(n) = min {k : (Sum_{c<=k, c composite} c)/(Sum_{p<=k, p prime} p) >= n}.

a(n) = min {k>1 : k(k+1)-1>=2*A034387(k)*(n+1)}. - Chai Wah Wu, Sep 10 2023

a(17)-a(21) from Chai Wah Wu, Sep 10 2023

A383641 a(n) is the difference between the sum of even composites and the sum of the odd composites in the first n positive integers.

Original entry on oeis.org

0, 0, 0, 4, 4, 10, 10, 18, 9, 19, 19, 31, 31, 45, 30, 46, 46, 64, 64, 84, 63, 85, 85, 109, 84, 110, 83, 111, 111, 141, 141, 173, 140, 174, 139, 175, 175, 213, 174, 214, 214, 256, 256, 300, 255, 301, 301, 349, 300, 350, 299, 351, 351, 405, 350, 406, 349, 407, 407

Offset: 1

Felix Huber, May 08 2025

nonn

			Of the first 9 positive integers, 4, 6, and 8 are even composites and 9 is an odd composite, so a(9) = 4 + 6 + 8 - 9 = 9.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Composite Number

Cf. A000720, A002808, A004526, A010701, A028552, A034387, A066247, A071904, A101256, A193356, A262044, A383259.

A383641:=n->`if`(n=1,0,floor((n-2)/2)-n*(n mod 2)+add(ithprime(i),i=2..NumberTheory:-pi(n)));seq(A383641(n),n=1..59);

lim=59;cn=Select[Range[lim],CompositeQ];a[n_]:=Total[Select[cn,EvenQ[#]&&#<=n&]]-Total[Select[cn,OddQ[#]&&#<=n&]];Array[a,lim] (* James C. McMahon, May 14 2025 *)

a(n) = floor((n-2)/2) - n*(n mod 2) + Sum_{i=2..pi(n)} prime(i) for n > 1.
a(n) = A004526(n) - A193356(n) - A010701(n) + A034387(A000720(n)) for n > 1.
a(n) = Sum_{i=1..n} ((-1)^i*i*A066247(i)).

A073700 a(1) = 1, a(n) = Floor[(Sum of composite numbers up to n)/(Sum of primes up to n)].

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Amarnath Murthy, Aug 12 2002

nonn

Though the sequence is not monotonically increasing the average value increases and a derived sequence could be the smallest value of k for which a(k) = n.

Note 1 is neither composite nor prime.

			a(10) = floor((4+6+8+9+10)/(2+3+5+7)) = floor(37/17) = 2.

Cf. A101256, A034387.

a := 0:b := 0:for i from 2 to 300 do if isprime(i) then a := a+i: else b := b+i:fi: c[i] := floor(b/a):od:c[1] := 1:seq(c[j],j=1..300);

Module[{nn=110,pr,comp},pr=Prime[Range[PrimePi[nn]]];comp=Complement[Range[ 2,nn], pr]; Join[{1}, Table[Floor[Total[Select[comp,#<=n&]]/Total[Select[pr,#<=n&]]],{n,2,nn}]]] (* Harvey P. Dale, Feb 22 2013 *)
Join[{1}, Table[t1 = Select[x = Range[n], PrimeQ]; Floor[Divide @@ Plus @@@ {Rest[Complement[x, t1]], t1}], {n, 2, 105}]] (* Jayanta Basu, Jul 07 2013 *)

a(n) = floor(A101256(n)/A034387(n)). - Jason Yuen, Aug 20 2024

More terms from Sascha Kurz, Aug 15 2002

cp's OEIS Frontend

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

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A364879 a(n) is the smallest number k such that (sum of composites <= k) / (sum of primes <= k) >= n.

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A383641 a(n) is the difference between the sum of even composites and the sum of the odd composites in the first n positive integers.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A073700 a(1) = 1, a(n) = Floor[(Sum of composite numbers up to n)/(Sum of primes up to n)].

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Programs

Maple

Mathematica

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