A034387 -id:A034387 - cp's OEIS frontend

A140235 Partial sum of non-semiprimes A100959.

1, 3, 6, 11, 18, 26, 37, 49, 62, 78, 95, 113, 132, 152, 175, 199, 226, 254, 283, 313, 344, 376, 412, 449, 489, 530, 572, 615, 659, 704, 751, 799, 849, 901, 954, 1008, 1064, 1123, 1183, 1244, 1307, 1371, 1437, 1504, 1572, 1642, 1713, 1785, 1858, 1933, 2009

Offset: 1

Jonathan Vos Post, May 13 2008

This is to semiprimes A001358 as A051352 is to primes A000040. Equivalently, this is to non-semiprimes A100959 as A051349 is to nonprimes A018252.

			a(5) = 18 = 1 + 2 + 3 + 5 + 7.

Cf. A000217, A001358, A018252, A034387, A051349, A051352, A062198, A072000, A100959, A140234.

Accumulate[Select[Range[100],PrimeOmega[#]!=2&]] (* Harvey P. Dale, Aug 22 2021 *)

a(n) = Sum{k=1..n} A100959(k).

Corrected and edited by Giovanni Resta, Jun 20 2016

A228357 Numbers n such that sum of all primes <=n is not prime.

Original entry on oeis.org

1, 5, 6, 11, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 41, 42, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Vincenzo Librandi, Aug 21 2013

			6 is in the sequence since 2+3+5=10 is not prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A034387, A228102.

[n: n in [1..120] | not IsPrime(s) where s is &+PrimesUpTo(n)];

t = {}; s = 0; Do[If [PrimeQ[n], s+ = n]; If[!PrimeQ[s], AppendTo[t, n]], {n, 120}]; t
Position[Accumulate[Table[If[PrimeQ[n],n,0],{n,100}]],?(!PrimeQ[ #]&)]// Flatten//Rest (* _Harvey P. Dale, Jul 02 2018 *)

A276355 Sum of primes between 100n and 100n + 99.

Original entry on oeis.org

1060, 3167, 4048, 5612, 7649, 7760, 10316, 10466, 12719, 13330, 16826, 13780, 18775, 14759, 24773, 18666, 24679, 21022, 22230, 25413, 28750, 21398, 33781, 35381, 24452, 28057, 39905, 38474, 34168, 32407, 36560, 31544, 35669, 50157, 38009, 49688, 47439, 44994

Offset: 0

Bhushan Bade, Aug 31 2016

The first occurrence of 0 in this sequence is as a(16718). - Robert Israel, Dec 28 2022

			Sum of primes in first interval of one hundred numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 is equal to first term i.e 1060.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A034387, A038822, A181098.

R:= NULL: m:= 0: p:= 0: s:= 0:
while m <= 100 do
  p:= nextprime(p);
  r:= floor(p/100);
  if r = m then
    s:= s + p;
  else
    R:= R, s;
    if m < r-1 then R:= R, 0$(r-1-m) fi;
    s:= p;
    m:= r;
  fi
od:
R;

Table[Total@ Select[Range[#, # + 99] &[100 n], PrimeQ], {n, 0, 37}] (* Michael De Vlieger, Sep 01 2016 *)

a(n) = A034387(100*(n+1)) - A034387(100*n). - Robert Israel, Aug 31 2016

Definition by Omar E. Pol, Aug 31 2016

A334122 a(n) is the sum of all primes <= n, mod n.

Original entry on oeis.org

0, 0, 2, 1, 0, 4, 3, 1, 8, 7, 6, 4, 2, 13, 11, 9, 7, 4, 1, 17, 14, 11, 8, 4, 0, 22, 19, 16, 13, 9, 5, 0, 28, 24, 20, 16, 12, 7, 2, 37, 33, 28, 23, 17, 11, 5, 46, 40, 34, 28, 22, 16, 10, 3, 51, 45, 39, 33, 27, 20, 13, 5, 60, 53, 46, 39, 32, 24, 16, 8, 0, 63, 55

Offset: 1

Christoph Schreier, Apr 15 2020

			a(7) = (2+3+5+7) mod 7 = 17 mod 7 = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720, A034387, A090396.

b:= proc(n) b(n):= `if`(n<2, 0, b(n-1)+`if`(isprime(n), n, 0)) end:
a:= n-> irem(b(n), n):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 15 2020

Mod[Accumulate[(# * Boole @ PrimeQ[#]) & /@ (r = Range[100])], r] (* Amiram Eldar, Apr 15 2020 *)

a(n) = my(np=primepi(n)); vecsum(primes(np)) % n; \\ Michel Marcus, Apr 16 2020

from sympy import primerange
a = lambda n: sum(primerange(n + 1)) % n # David Radcliffe, May 10 2025

a(n) = A034387(n) mod n.

A349214 a(n) = Sum_{k=1..n} k^c(k), where c is the prime characteristic (A010051).

Original entry on oeis.org

1, 3, 6, 7, 12, 13, 20, 21, 22, 23, 34, 35, 48, 49, 50, 51, 68, 69, 88, 89, 90, 91, 114, 115, 116, 117, 118, 119, 148, 149, 180, 181, 182, 183, 184, 185, 222, 223, 224, 225, 266, 267, 310, 311, 312, 313, 360, 361, 362, 363, 364, 365, 418, 419, 420, 421, 422, 423, 482, 483, 544

Offset: 1

Wesley Ivan Hurt, Nov 10 2021

nonn

For k in 1 <= k <= n, add k if k is prime, otherwise add 1. For example a(6) = 1 + 2 + 3 + 1 + 5 + 1 = 13.

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

Partial sums of A089026.

Cf. A010051, A034387, A062298.

a[n_] := Sum[k^Boole[PrimeQ[k]], {k, 1, n}]; Array[a, 60] (* Amiram Eldar, Nov 11 2021 *)

a(n) = sum(k=1, n, if (isprime(k), k, 1)); \\ Michel Marcus, Nov 11 2021

from sympy import primerange
def A349214(n):
    p = list(primerange(2,n+1))
    return n-len(p)+sum(p) # Chai Wah Wu, Nov 11 2021

a(n) = A034387(n) + A062298(n). - Wesley Ivan Hurt, Nov 23 2021

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

A082089 a(n)-th prime is the fixed point if function A008472[=sum of prime factors with no repetition] is iterated when started at factorial of n-th prime.

Original entry on oeis.org

1, 3, 4, 7, 2, 13, 11, 3, 4, 3, 4, 45, 1, 60, 14, 4, 3, 3, 21, 1, 4, 4, 6, 3, 4, 3, 2, 4, 6, 2, 4, 4, 4, 4, 105, 4, 4, 3, 4, 4, 3, 4, 3, 4, 1, 4, 8, 2, 2, 19, 3, 1, 20, 14, 4, 20, 52, 4, 4, 977, 1, 3, 65, 1108, 1, 2, 46, 3, 3, 1, 3, 1, 2, 4, 829, 2, 25, 3, 8, 25, 4, 378, 3, 3, 29, 3, 6, 8, 1, 1, 28

Offset: 2

Labos Elemer, Apr 09 2003

nonn

a(n) < n holds usually, except few large values arising unexpectedly.

			n=100, p(100)=541, starts at factorial of 100th prime and ends in 24133, the 2687th prime, so a(100)=2687;
n=99, initial value=523!, fixed point is 19, the 8th prime, a(99)=8.

Cf. A008472, A034387, A007504, A075860, A082087, A082088.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Table[PrimePi[FixedPoint[sopf, Prime[w]! ]], {w, 2, 100}]

a(n) = A000720(A082087(A000142(A000040(n)))) = pi(A082087(p(n)!)).

A115333 Sum of primes that do not divide n and are less than the largest prime dividing n.

Original entry on oeis.org

0, 0, 2, 0, 5, 0, 10, 0, 2, 3, 17, 0, 28, 8, 2, 0, 41, 0, 58, 3, 7, 15, 77, 0, 5, 26, 2, 8, 100, 0, 129, 0, 14, 39, 5, 0, 160, 56, 25, 3, 197, 5, 238, 15, 2, 75, 281, 0, 10, 3, 38, 26, 328, 0, 12, 8, 55, 98, 381, 0, 440, 127, 7, 0, 23, 12, 501, 39, 74, 3, 568, 0, 639, 158, 2, 56, 10

Offset: 1

Leroy Quet, Mar 05 2006

nonn

When n is prime, n = largest prime dividing n; hence a(n) is the sum of all primes less than n = A034387(n)-n. a(n) = SUM{p such that p is in A000040 AND NOT(p|n) AND p < A006530(n)}. - Jonathan Vos Post, Mar 08 2006

The zeros give A055932: All prime divisors are consecutive primes starting at 2. - Robert G. Wilson v, May 01 2006

			The primes < 7 and coprime to 7 are 2, 3 and 5. So a(7) = 2+3+5 = 10.

Cf. A000040, A006530, A034387, A066911, A083720.

f[n_] := Plus @@ Complement[Prime@ Range@ PrimePi[ Max[First /@ FactorInteger@n] - 1], First /@ FactorInteger@n]; Array[f, 77] (* Hans Havermann, Mar 06 2006 *)

More terms from Hans Havermann, Mar 06 2006

cp's OEIS Frontend

A140235 Partial sum of non-semiprimes A100959.

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A228357 Numbers n such that sum of all primes <=n is not prime.

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A276355 Sum of primes between 100*n and 100*n + 99.

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A334122 a(n) is the sum of all primes <= n, mod n.

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A349214 a(n) = Sum_{k=1..n} k^c(k), where c is the prime characteristic (A010051).

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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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Maple

A276355 Sum of primes between 100n and 100n + 99.