A378678 a(n) = numerator(Sum_{k=1..n} 1/P(k)), where P(k) = A006530(k) is the greatest prime dividing k for k >= 2, and P(1) = 1.

1, 3, 11, 7, 38, 43, 316, 737, 269, 283, 3183, 10319, 136457, 140747, 146753, 80884, 1390043, 1475128, 28282687, 29252656, 29945491, 30386386, 703736723, 246972956, 50881877, 51453910, 161798159, 164985200, 4806880087, 24681369758, 768357309113, 1636994863291

Offset

1,2

References

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory, Vol. 1, Springer, 2006, Chapter IV, p. 122.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Paul Erdős and Aleksandar Ivić, Estimates for sums involving the largest prime factor of an integer and certain related additive functions, Studia Sci. Math. Hungar, Vol. 15, No. 1-3 (1980), pp. 183-199.

Paul Erdős and Aleksandar Ivić, On sums involving reciprocals of certain arithmetical functions, Publ. Inst. Math.(Beograd)(NS), Vol. 32, No. 46 (1982), pp. 49-56.

Paul Erdős, Aleksandar Ivić, and Carl Pomerance, On sums involving reciprocals of the largest prime factor of an integer, Glasnik Matematicki, Vol. 21, No. 41 (1986), pp. 283-300.

Aleksandar Ivić, Sum of reciprocals of the largest prime factor of an integer, Archiv der Mathematik, Vol. 36 (1981), pp. 57-61.

Aleksandar Ivić, On sums involving reciprocals of the largest prime factor of an integer II, Acta Arithmetica, Vol. 71, No. 3 (1995), pp. 229-251

Aleksandar Ivić and Carl Pomerance, Estimates for certain sums involving the largest prime factor of an integer, Coll. Math. Soc. J. Bolyai, Vol. 34 (1981), pp. 769-789.

Formula

a(n)/A378679(n) = n * exp(-sqrt(2*log(n)*log(log(n))) + O(sqrt(log(n)*log(log(log(n)))))) (Ivić, 1981).

Example

Fractions begin: 1, 3/2, 11/6, 7/3, 38/15, 43/15, 316/105, 737/210, 269/70, 283/70, 3183/770, 10319/2310, ...

Mathematica

Numerator@ Accumulate[Table[1/FactorInteger[n][[-1, 1]], {n, 1, 33}]]

Prog

(PARI) lista(nmax) = {my(s = 1); print1(1, ", "); for(n = 2, nmax, f = factor(n); s += 1/f[#f~, 1]; print1(numerator(s), ", ")); }

Crossrefs

Cf. A006530, A057158, A378679 (denominators).

Sequence in context: A322364 A250034 A379367 * A006495 A112286 A126261

Adjacent sequences: A378675 A378676 A378677 * A378679 A378680 A378681

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 03 2024

Status

approved