A068885 -id:A068885 - cp's OEIS frontend

A069947 Denominator of Sum_{k=1..n} k/phi(k).

1, 1, 2, 2, 4, 4, 12, 12, 12, 12, 60, 60, 10, 30, 120, 120, 240, 240, 720, 720, 720, 720, 7920, 7920, 7920, 7920, 7920, 7920, 55440, 55440, 11088, 11088, 55440, 55440, 55440, 55440, 18480, 55440, 55440, 55440, 55440, 55440, 55440, 11088, 11088, 11088, 255024, 255024

Offset: 1

N. J. A. Sloane, Jun 28 2002

			1, 3, 9/2, 13/2, 31/4, 43/4, 143/12, 167/12, 185/12, ...

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter I, p. 29, section I.27.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A068885 (numerators), A028415, A048049.

map(denom, ListTools:-PartialSums([seq(n/numtheory:-phi(n),n=1..100)]));
# Robert Israel, May 01 2018

Accumulate[Table[n/EulerPhi[n],{n,50}]]//Denominator (* Harvey P. Dale, May 24 2021 *)

a(n) = denominator(sum(k = 1, n, k/eulerphi(k))); \\ Amiram Eldar, Apr 25 2024

A385562 Numbers m such that (1/m) * Sum_{k=1..m} k/phi(k) sets a record value, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 22, 24, 30, 42, 60, 66, 72, 78, 84, 90, 114, 120, 150, 156, 180, 198, 210, 300, 330, 390, 420, 510, 546, 570, 600, 630, 750, 780, 840, 966, 990, 1122, 1170, 1200, 1260, 1410, 1470, 1560, 1596, 1620, 1650, 1680, 1806, 1830, 1890, 1980, 2100

Offset: 1

Amiram Eldar, Jul 03 2025

nonn

Limit_{m->oo} (1/m) * Sum_{k=1..m} k/phi(k) = zeta(2)*zeta(3)/zeta(6) (A082695) (Sitaramachandrarao, 1985; Sándor et al., 2005). This sequence is infinite if this mean converges to the limit only from below.

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 29.

Amiram Eldar, Table of n, a(n) for n = 1..1752 (terms below 10^10)
R. Sitaramachandrarao, On an error term of Landau. II, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp. 579-588.

Cf. A000010, A068885, A069947, A082695, A330899, A385561.

seq[lim_] := Module[{s = {}, sum = 0, rm = 0, r}, Do[sum += k/EulerPhi[k]; r = sum/k; If[r > rm, rm = r; AppendTo[s, k]], {k, 1, lim}]; s]; seq[2500]

list(lim) = {my(sm = 0, rm = 0, r); for(k = 1, lim, sm += k/eulerphi(k); r = sm/k; if(r > rm, rm = r; print1(k, ", ")));}

cp's OEIS Frontend

A069947 Denominator of Sum_{k=1..n} k/phi(k).

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