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

Author

Amiram Eldar, Dec 03 2024

Keywords

• nonn
• easy
• frac

Examples

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

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.

Crossrefs

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

Programs

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]}]]

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), ", "));}

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.