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A002098
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G.f.: 1/Product_{k>=1} (1-prime(k)*x^prime(k)).
(Formerly M0597 N0215)
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8
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1, 0, 2, 3, 4, 11, 17, 29, 49, 85, 144, 226, 404, 603, 1025, 1679, 2558, 4201, 6677, 10190, 16599, 25681, 39643, 61830, 96771, 147114, 228338, 352725, 533291, 818624, 1263259, 1885918, 2900270, 4396577, 6595481, 10040029, 15166064, 22642064
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OFFSET
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0,3
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COMMENTS
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a(n) is sum of all numbers k for which A001414(k), the sum of prime factors with repetition, equals n. See Havermann's link. - J. M. Bergot, Jun 14 2013
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REFERENCES
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S.M. Kerawala, On a Pair of Arithmetic Functions Analogous to Chawla's Pair, J. Natural Sciences and Mathematics, 9 (1969), circa p. 103.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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MAPLE
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b:= proc(n, i) option remember;
if n<0 then 0
elif n=0 then 1
elif i=0 then 0
else b(n, i-1) +b(n-ithprime(i), i) *ithprime(i)
fi
end:
a:= n-> b(n, numtheory[pi](n)):
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MATHEMATICA
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b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i==0, 0, True, b[n, i-1] + b[n - Prime[i], i]*Prime[i]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
With[{nn=40}, CoefficientList[Series[1/Product[1-Prime[k]x^Prime[k], {k, nn}], {x, 0, nn}], x]] (* Harvey P. Dale, Jun 20 2021 *)
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PROG
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(PARI) my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-isprime(k)*k*x^k)) \\ Seiichi Manyama, Feb 27 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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