A074467 -id:A074467 - cp's OEIS frontend

A074468 Least number m such that the Sigma-Harmonic sequence Sum_{k=1..m} 1/sigma(k) >= n.

1, 7, 29, 129, 571, 2525, 11167, 49372, 218295, 965177, 4267457, 18868240, 83424514, 368855252, 1630865929, 7210751807, 31881800153

Offset: 1

Labos Elemer, Aug 29 2002

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 129, p. 44, Ellipses, Paris, 2008.

Cf. A000203 (sigma), A002387, A004080, A046024, A074631, A074633, A074467, A212717, A308039.

{s=0, s1=0}; Do[s=s+(1/DivisorSigma[1, n]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 1, 1000000}]

Limit_{n->oo} a(n+1)/a(n) = exp(1/c) = 4.42142525588146107878... where c = A308039. - Amiram Eldar, May 05 2024

2 more terms from Lekraj Beedassy, Jul 14 2008
a(11)-a(15) from Donovan Johnson, Aug 22 2011
a(16)-a(17) from Amiram Eldar, May 05 2024

A146323 a(n) = floor(Sum_{i=1..n} (1/phi(i))).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9

Offset: 1

Ctibor O. Zizka, Oct 30 2008

nonn

Looking on the number of 1's, 2's, ..., k's in this sequence we obtain the sequence (1,2,4,5,9,16,25,42,72,...). Limit_{k->oo} (number of (k+1)'s / number of(k's)) = sqrt(e).

The limit above is wrong. The correct limit is exp(zeta(6)/(zeta(2)*zeta(3))) = exp(1/A082695) = 1.672818789624... . - Amiram Eldar, Jul 04 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A028415, A048049, A074467, A082695.

IntegerPart[Accumulate[1/EulerPhi[Range[110]]]] (* Harvey P. Dale, Dec 19 2015 *)

list(lim) = {my(s = 0); for(k = 1, lim, s += 1/eulerphi(k); print1(floor(s), ", "));} \\ Amiram Eldar, Jul 04 2025

a(n) = floor(A028415(n)/A048049(n)). - Amiram Eldar, Jul 04 2025

cp's OEIS Frontend

A074468 Least number m such that the Sigma-Harmonic sequence Sum_{k=1..m} 1/sigma(k) >= n.

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