A378680 a(n) = numerator(Sum_{k=1..n} 1/P_2(k)), where P_2(k) = A087040(k) is the second largest prime dividing the k-th composite number.

1, 1, 3, 11, 7, 17, 10, 11, 25, 9, 5, 16, 35, 19, 98, 211, 221, 118, 41, 87, 271, 143, 146, 151, 317, 109, 57, 176, 367, 377, 196, 407, 2879, 2921, 997, 516, 1583, 1604, 3313, 3383, 1744, 593, 1221, 3733, 1919, 388, 395, 811, 275, 1389, 4237, 2171, 2192, 4489

Offset

1,3

Links

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

Jean-Marie De Koninck, Sur les plus grands facteurs premiers d'un entier, Monatshefte für Mathematik, Vol. 116, No. 1 (1993), pp. 13-37; alternative link; author's copy.

Formula

a(n)/A378681(n) = Sum_{k=1..m} c_k * n/log(n)^k + O(n/log(n)^(m+1)) for any integer m >= 1, where c_k are constants. c_1 = Sum_{k>=1} (1/k)*Sum_{p prime > P(k)} 1/p^2 = Sum_{p prime} (1/p^2)*Product_{primes q < p} (1/(1-1/q)) = 1.254435359..., where P(k) = A006530(k) is the greatest prime dividing k for k >= 2, and P(1) = 1.

Example

Fractions begin: 1/2, 1, 3/2, 11/6, 7/3, 17/6, 10/3, 11/3, 25/6, 9/2, 5, 16/3, ...

Mathematica

p2[c_] := Module[{f = FactorInteger[c]}, If[f[[-1, 2]] > 1, f[[-1, 1]], f[[-2, 1]]]]; Numerator@ Accumulate[Table[1/p2[c], {c, Select[Range[100], CompositeQ]}]]

Prog

(PARI) lista(nmax) = {my(s = 0); forcomposite(n = 1, nmax, f = factor(n); s += if(f[#f~, 2] > 1, 1/f[#f~, 1], 1/f[#f~ - 1, 1]); print1(numerator(s), ", ")); }

Crossrefs

Cf. A006530, A087039, A087040, A378681 (denominators).

Sequence in context: A288902 A288832 A164808 * A355895 A153285 A083557

Adjacent sequences: A378677 A378678 A378679 * A378681 A378685 A378686

Keyword

nonn,easy,frac,new

Author

Amiram Eldar, Dec 03 2024

Status

approved