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).

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

Author

Amiram Eldar, Jul 03 2025

Keywords

• nonn

Comments

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.

References

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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]

PARI

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, ", ")));}