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

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

Offset: 1

Author

Max Alekseyev, Feb 10 2018

Keywords

• nonn
• more
• nice

Examples

			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)
		

Crossrefs

Intersection of A007504 and A053767.

Cf. A066527, A154587.

Programs

Mathematica

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 *)