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%I #34 Mar 11 2024 01:53:57
%S 0,10,1988,14697,83292,1503397,18859052,93952013,89171409882,
%T 9646383703961,209456854921713,3950430820867201,
%U 13113506646374409451778
%N Numbers that can be expressed both as the sum of first primes and as the sum of first composites.
%e From _Jon E. Schoenfield_, Feb 10 2018: (Start)
%e 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).
%e 1988 is in the sequence because Sum_{i=1..33} prime(i) = A007504(33) = Sum_{i=1..51} composite(i) = A053767(51) = 1988.
%e a(n) = A007504(j)
%e n j k = A053767(k)
%e == ======== ======== =================
%e 1 0 0 0
%e 2 3 2 10
%e 3 33 51 1988
%e 4 80 147 14697
%e 5 175 361 83292
%e 6 660 1582 1503397
%e 7 2143 5699 18859052
%e 8 4556 12821 93952013
%e 9 118785 403341 89171409882
%e 10 1131142 4229425 9646383703961
%e 11 5012372 19786181 209456854921713
%e 12 20840220 86192660 3950430820867201 (End)
%t 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 *)
%t 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 *)
%Y Intersection of A007504 and A053767.
%Y Cf. A066527, A154587.
%K nonn,more,nice
%O 1,2
%A _Max Alekseyev_, Feb 10 2018