A069947 -id:A069947 - cp's OEIS frontend

A068885 Numerator of Sum_{k=1..n} k/phi(k).

1, 3, 9, 13, 31, 43, 143, 167, 185, 215, 1141, 1321, 231, 763, 3277, 3517, 7289, 8009, 24787, 26587, 27847, 29431, 332021, 355781, 365681, 382841, 394721, 413201, 2949827, 3157727, 643003, 665179, 3417371, 3535181, 3616031, 3782351, 1279777, 3956371, 4046461

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, Section I.27, page 29.

Amiram Eldar, Table of n, a(n) for n = 1..1000
R. Sitaramachandrarao, On an error term of Landau - II, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp. 579-588.

Cf. A069947 (denominators), A000010, A028415, A048049, A082695.

Numerator @ Accumulate @ Table[k/EulerPhi[k], {k, 1, 40}] (* Amiram Eldar, Sep 18 2022 *)

a(n) = numerator(sum(k=1, n, k/eulerphi(k))); \\ Michel Marcus, Sep 18 2022

a(n)/A069947(n) ~ c * n - log(n)/2 + O(log(n)^(2/3)), where c = zeta(2)*zeta(3)/zeta(6) (A082695) (Sitaramachandrarao, 1985). - Amiram Eldar, Sep 18 2022

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

A068885 Numerator of Sum_{k=1..n} k/phi(k).

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