A145191 Numbers m such that Sum_{i=1..m} omega(i)^2 is divisible by m, where omega is A001221.

1, 20, 68, 903, 3876, 3890, 19096, 19122, 19127, 110990, 111004, 111007, 111010, 111013, 774276, 774277, 774278, 774279, 774303, 774313, 774314, 774315, 6615593, 70607550, 70607559, 959878582, 959878737, 959878753, 959878836, 959878846, 959878888, 959878902, 959878914

Offset

1,2

Comments

If for some m is c square, then we have RootMeanSquare(omega(1),...,omega(n)) = c.

Links

Table of n, a(n) for n=1..33.

Mathematica

With[{max = 10^5}, Position[Accumulate[PrimeNu[Range[max]]^2]/Range[max], _?IntegerQ] // Flatten] (* Amiram Eldar, Sep 22 2024 *)

Prog

(PARI) isok(m) = !frac(sum(i=1, m, omega(i)^2)/m); \\ Michel Marcus, Mar 15 2022
(PARI) lista(nn) = {my(v = vector(nn, k, omega(k)^2)); print1(1, ", "); for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/n), print1(n, ", ")); ); } \\ Michel Marcus, Mar 16 2022
(PARI) listaa(nn) = {my(v = vector(nn, k, omega(k)^2)); print1(1, ", "); for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/n), print1(n, ", ")); ); for (m=1, 100, last = v[nn]; v = vector(nn, k, omega(k+m*nn)^2); v[1] += last; for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/(m*nn+n)), print1(n+m*nn, ", ")); ); ); } \\ Michel Marcus, Mar 16 2022

Crossrefs

Cf. A001221, A063971, A013939, A140480.

Sequence in context: A235281 A303620 A221521 * A044158 A044539 A297597

Adjacent sequences: A145188 A145189 A145190 * A145192 A145193 A145194

Keyword

nonn

Author

Ctibor O. Zizka, Oct 03 2008

Extensions

a(7)-a(9) from Michel Marcus, Mar 15 2022

a(10)-a(25) from Michel Marcus, Mar 16 2022

a(26)-a(33) from Amiram Eldar, Sep 22 2024

Status

approved