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

Author

Ctibor O. Zizka, Oct 03 2008

Keywords

• nonn

Comments

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

Crossrefs

Cf. A001221, A063971, A013939, A140480.

Programs

Mathematica

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

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

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