A028236 If n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j.

1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 7, 1, 9, 8, 1, 1, 11, 1, 9, 10, 13, 1, 11, 1, 15, 1, 11, 1, 31, 1, 1, 14, 19, 12, 13, 1, 21, 16, 13, 1, 41, 1, 15, 14, 25, 1, 19, 1, 27, 20, 17, 1, 29, 16, 15, 22, 31, 1, 47, 1, 33, 16, 1, 18, 61, 1, 21, 26, 59, 1, 17, 1, 39, 28, 23, 18, 71, 1, 21, 1, 43

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with 1, 1/2, 1/3, 1/4, 1/5, 5/6, 1/7, 1/8, 1/9, 7/10, 1/11, 7/12, ...
		

Links

• Klaus Brockhaus, Table of n, a(n) for n = 1..10000

Crossrefs

Denominator is n (A000027).

Cf. A001221, A141809, A179119.

Programs

Haskell

a028236 n = sum $ map (div n) $ a141809_row n
-- Reinhard Zumkeller, Nov 10 2013

Magma

a028236:=func< k | k eq 1 select 1 else Numerator(&+[ f[i, 1]^-f[i,2]: i in [1..#f] ]) where f is Factorization(k) >; [ a028236(n):n in [1..82] ]; // Klaus Brockhaus, Nov 06 2010

Mathematica

a[n_] := n * Total[1/Power @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Sep 29 2023 *)

Formula

Fraction is additive with a(p^e) = 1/p^e.

a(n) = Sum_{k=1..A001221(n)} n/A141809(n,k). - Reinhard Zumkeller, Nov 10 2013

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/k = Sum_{p prime} 1/(p*(p+1)) = 0.330229... (A179119). - Amiram Eldar, Sep 29 2023

Extensions

More terms from Erich Friedman