A053767 -id:A053767 - cp's OEIS frontend

A023539 Convolution of natural numbers with composite numbers.

4, 14, 32, 59, 96, 145, 208, 286, 380, 492, 624, 777, 952, 1151, 1375, 1625, 1902, 2207, 2542, 2909, 3309, 3743, 4212, 4717, 5260, 5842, 6464, 7128, 7836, 8589, 9388, 10235, 11131, 12077, 13074, 14123, 15226, 16384, 17598, 18869, 20198

Offset: 1

Clark Kimberling

nonn

			a(1) = 1*4 = 4;
a(2) = 1*6 + 2*4 = 14;
a(3) = 1*8 + 2*6 + 3*4 = 32;

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002808.

First differences are in A053767.

lista(nn) = {my(vc = []); forcomposite(n=2, nn, vc = concat(vc, n); print1(sum(k=1, #vc, (#vc-k+1)*vc[k]), ", "););} \\ Michel Marcus, Feb 11 2018

A227249 Number of consecutive composites beginning with the first, to be added to obtain a power.

Original entry on oeis.org

1, 4, 6, 21, 80, 4151, 6982, 269563, 779693, 834365, 16176645, 19770092, 41049539, 228612936, 1950787140, 2404785364, 3095996836, 5236785750

Offset: 1

Robin Garcia, Jul 04 2013

All powers are squares with the exception of 3^3 for a(2) and 6^9 for a(6). I conjecture these are the only nonsquare powers.

a(19) > 10^10. - Zak Seidov, Jul 06 2013

			Considering 1 not to be prime and not to be composite, first composite is 4 which is 2^2. And the sum of the first four composites is 4 + 6 + 8 + 9 = 27 = 3^3.

Cf. A001597, A002808, A053767, A141092.

# see A001597 for isA001597
for n from 1 do
    if isA001597(A053767(n) ) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Jul 08 2013

n=10^7;v=vector(n);i=0;for(a=2,n,if(isprime(a),next,i++;v[i]=a));k=0;for(j=1,i,k=k+v[j];if(ispower(k,,&n),print1([k,n,j]," ")))

{n: A053767(n) in A001597}. - Zak Seidov, Jul 06 2013

a(11) - a(18) from Zak Seidov, Jul 06 2013

A294174 Numbers that can be expressed both as the sum of first primes and as the sum of first composites.

Original entry on oeis.org

0, 10, 1988, 14697, 83292, 1503397, 18859052, 93952013, 89171409882, 9646383703961, 209456854921713, 3950430820867201, 13113506646374409451778

Offset: 1

Max Alekseyev, Feb 10 2018

			From _Jon E. Schoenfield_, Feb 10 2018: (Start)
10 is in the sequence because prime(1) + prime(2) + prime(3) = 2 + 3 + 5 = 10 and composite(1) + composite(2) = 4 + 6 = 10 (where composite(i) is the i-th composite number).
1988 is in the sequence because Sum_{i=1..33} prime(i) = A007504(33) = Sum_{i=1..51} composite(i) = A053767(51) = 1988.
                          a(n) = A007504(j)
   n         j         k       = A053767(k)
  ==  ========  ========  =================
   1         0         0                  0
   2         3         2                 10
   3        33        51               1988
   4        80       147              14697
   5       175       361              83292
   6       660      1582            1503397
   7      2143      5699           18859052
   8      4556     12821           93952013
   9    118785    403341        89171409882
  10   1131142   4229425      9646383703961
  11   5012372  19786181    209456854921713
  12  20840220  86192660   3950430820867201 (End)

Intersection of A007504 and A053767.

Cf. A066527, A154587.

nextComposite[n_] := Block[{k = n + 1}, While[PrimeQ@k, k++]; k]; c = sc = 4; p = sp = 2; lst = {0}; While[p < 1000000000, If[ sc == sp, AppendTo[lst, sc]; c = nextComposite@c; sc += c]; While[ sp < sc, p = NextPrime@ p; sp += p]; While[ sc < sp, c = nextComposite@ c; sc += c]]; lst (* Robert G. Wilson v, Feb 11 2018 *)
Module[{pr=Accumulate[Prime[Range[5*10^7]]],co=Accumulate[Select[ Range[ 11*10^7], CompositeQ]]},Join[ {0},Intersection[pr,co]]] (* The program generates the first 12 terms of the sequence; to generate the 13th term increase the Range specifications substantially, but the program will take a long time to run. *) (* Harvey P. Dale, Sep 17 2019 *)

A330578 a(n) is the remainder when the sum of the first n composite numbers is divided by the n-th composite number.

Original entry on oeis.org

0, 4, 2, 0, 7, 1, 7, 3, 14, 4, 12, 6, 21, 7, 24, 16, 7, 25, 5, 15, 4, 26, 14, 1, 11, 36, 22, 34, 4, 33, 17, 31, 14, 46, 28, 9, 23, 3, 38, 17, 53, 9, 25, 2, 42, 18, 59, 9, 52, 26, 44, 64, 37, 9, 57, 28, 48, 18, 69, 7, 60, 28, 82, 49, 71, 37, 2, 59, 23, 81, 44

Offset: 1

Rémy Sigrist, Dec 18 2019

nonn

			a(3) = (4 + 6 + 8) mod 8 = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A002808, A053767, A071089 (prime variant), A330579 (positions of zeros).

s=0; forcomposite (c=4, 96, s+=c; print1 (s%c", "))

a(n) = A053767(n) mod A002808(n).

A065022 Composite n such that the sums of the composite numbers up to n, +/- 1, are twin primes.

Original entry on oeis.org

4, 8, 290, 340, 352, 412, 489, 610, 774, 785, 1227, 1295, 1306, 1795, 1853, 1918, 1945, 2014, 2266, 2502, 2885, 3063, 3133, 3178, 3265, 3482, 3486, 3680, 3760, 3843, 3973, 3995, 4124, 4794, 5677, 5769, 5965, 6123, 7555, 7653, 7696, 7765, 7786, 8023

Offset: 1

Robert G. Wilson v, Nov 01 2001

nonn

			4+6+8 = 18 and 18 +/-1 are twin primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014574, A053767.

Composite[n_Integer] := (k = n + PrimePi[n] + 1; While[k - PrimePi[k] - 1 != n, k++ ]; k); s = 0; Do[m = Composite[n]; s = s + m; If[ PrimeQ[s - 1] && PrimeQ[s + 1], Print[m]], {n, 1, 10^4} ]

is(n)=my(s); if(isprime(n),return(0)); forcomposite(k=4,n,s+=k);isprime(s-1)&&isprime(s+1) \\ Charles R Greathouse IV, Jan 02 2014

s=0;forcomposite(n=4,8023,s+=n;if(isprime(s-1) && isprime(s+1), print1(n", "))) \\ Charles R Greathouse IV, Jan 02 2014

New name from Charles R Greathouse IV, Jan 02 2014

A227314 Number of prime factors, with multiplicity, of the sum of the first n composite numbers.

Original entry on oeis.org

2, 2, 3, 3, 1, 2, 3, 3, 2, 5, 4, 3, 3, 1, 6, 4, 1, 2, 2, 1, 6, 3, 2, 2, 2, 3, 2, 4, 4, 2, 2, 3, 8, 3, 1, 1, 1, 3, 2, 2, 2, 2, 1, 2, 3, 2, 1, 1, 3, 3, 4, 2, 1, 1, 3, 7, 7, 3, 2, 4, 4, 2, 1, 1, 2, 4, 2, 1, 3, 4, 4, 3, 1, 1, 1, 2, 5, 2, 3, 6, 2, 2, 3, 6, 5, 1, 5, 3, 2, 3, 2, 6, 7, 2, 2, 2, 6, 3, 2, 3

Offset: 1

Jonathan Vos Post, Jul 06 2013

			a(10) = 5 because A053767(10) = 4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 + 16 + 18 = 112 = 2^4 * 7, which has 5 prime factors, with multiplicity.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A002808, A053767.

A227314 := proc(n)
    A001222(A053767(n)) ;
end proc: # R. J. Mathar, Jul 06 2013

PrimeOmega[Accumulate[Select[Range[150], CompositeQ]]] (* Amiram Eldar, Oct 06 2024 *)

a(n) = A001222(A053767(n)).

A257392 Number of ways of representing n as the sum of one or more consecutive nonprime numbers (A018252).

Original entry on oeis.org

1, 2, 0, 0, 1, 2, 1, 0, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 3, 3, 1, 1, 2, 1, 2, 2, 1, 2, 1, 4, 2, 1, 2, 1, 1, 1, 4, 2, 0, 1, 4, 3, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 0, 2, 4, 3, 2, 1, 3, 2, 3, 1, 2, 1, 0, 3, 4, 2, 2, 3, 3, 1, 3, 2, 1, 2, 3, 2, 2, 1, 1, 4, 2, 2, 1, 4, 5

Offset: 1

Juri-Stepan Gerasimov, Apr 21 2015

nonn

			a(2) = 2 because n = 2 itself is already a nonprime number (sum of 1 term), and 1 can in addition be written as A018252(1) + A018252(2), a sum of 2 consecutive nonprime numbers.

Cf. A051349, A053767, A018252, A166037, A166039, A242667, A257393.

A298270 Triangular numbers that for some k are also the sum of the first k composites.

Original entry on oeis.org

0, 10, 78, 153, 946, 177310, 450775, 13595505, 150988753, 4478601403, 5409300078, 5589152128, 76060335351, 248156250265, 1793751529485, 176149383165876, 187718592284301, 233626949305596, 11362376565228270, 18886935830647605, 1943937379018997076

Offset: 1

Altug Alkan, Feb 15 2018

nonn

			10 is a term because 10 = 1 + 2 + 3 + 4 = 4 + 6.

Intersection of A000217 and A053767.

Cf. A066527, A154587, A294174.

Join[{0},Select[Accumulate[Select[Range[10^6],CompositeQ]],OddQ[Sqrt[8#+1]]&]] (* The program generates the first 14 terms of the sequence. *) (* Harvey P. Dale, Apr 20 2024 *)

lista(nn) = {my(s=0); forcomposite(n=0, nn, if(ispolygonal(s, 3), print1(s, ", ")); s += n; ); } \\ after Michel Marcus at A053767

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

A132996 a(n) = gcd(Sum_{k=1..n} c(k), Product_{j=1..n} c(j)), where c(k) is the k-th composite.

Original entry on oeis.org

4, 2, 6, 27, 1, 1, 63, 6, 2, 112, 12, 9, 175, 1, 224, 250, 1, 5, 5, 1, 400, 14, 7, 5, 3, 6, 2, 8, 12, 3, 17, 847, 896, 22, 1, 1, 1, 6, 2, 1, 3, 3, 1, 2, 6, 31, 1, 1, 26, 4, 28, 2, 1, 1, 10, 2368, 2448, 9, 7, 2695, 20, 2, 1, 1, 31, 18, 2, 1, 9, 3596, 52, 10, 1, 1, 1, 5, 4300, 2, 74, 4624

Offset: 1

Leroy Quet, Nov 22 2007

nonn

			The first 8 composites are 4,6,8,9,10,12,14,15. 4+6+8+9+10+12+14+15 = 78 = 2*3*13. So a(8) = gcd(2*3*13, 4*6*8*9*10*12*14*15) = 6.

Cf. A053767, A036691.

lim=80;c[n_]:=n-PrimePi[n]-1;i=0;Do[Until[c[i]==m,i++];Cmp[m]=i,{m,lim}];Table[GCD[Sum[Cmp[k],{k,n}],Product[Cmp[j],{j,n}]],{n,lim}] (* James C. McMahon, Mar 09 2025 *)

More terms from R. J. Mathar, Jan 13 2008

cp's OEIS Frontend

A023539 Convolution of natural numbers with composite numbers.

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PARI

A227249 Number of consecutive composites beginning with the first, to be added to obtain a power.

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Maple

PARI

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Extensions

A294174 Numbers that can be expressed both as the sum of first primes and as the sum of first composites.

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Mathematica

A330578 a(n) is the remainder when the sum of the first n composite numbers is divided by the n-th composite number.

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PARI

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A065022 Composite n such that the sums of the composite numbers up to n, +/- 1, are twin primes.

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Mathematica

PARI

PARI

Extensions

A227314 Number of prime factors, with multiplicity, of the sum of the first n composite numbers.

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Maple

Mathematica

Formula

A257392 Number of ways of representing n as the sum of one or more consecutive nonprime numbers (A018252).

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A298270 Triangular numbers that for some k are also the sum of the first k composites.

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Mathematica

PARI

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

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