A069811 a(n) = Sum_{k=1..n} omega(k)^2.

0, 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 20, 21, 25, 29, 30, 31, 35, 36, 40, 44, 48, 49, 53, 54, 58, 59, 63, 64, 73, 74, 75, 79, 83, 87, 91, 92, 96, 100, 104, 105, 114, 115, 119, 123, 127, 128, 132, 133, 137, 141, 145, 146, 150, 154, 158, 162, 166, 167, 176, 177, 181

Offset: 1

Author

Benoit Cloitre, Apr 30 2002

Keywords

• easy
• nonn

References

• G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Fifth edition, Oxford University Press, Chap. XXII, p. 357.

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Paul Turán, On a theorem of Hardy and Ramanujan, Journal of the London Mathematical Society, Vol. s1-9, No. 4 (1934), pp. 274-276.

Crossrefs

Equals (A074787(n)-1)/4 for n <= 29.

Cf. A001221, A346617.

Programs

Mathematica

Accumulate@((PrimeNu@Range@62)^2) (* Ivan Neretin, Mar 16 2017 *)

PARI

a(n) = sum(k=1, n, omega(k)^2); \\ Michel Marcus, Mar 16 2017

Formula

a(n) = Sum_{k=1..n} A001221(k)^2.

a(n) = n*log(log(n))^2 + O(n*log(log(n))) (Turán, 1934).

a(n) = Sum_{k>=1} k^2 * A346617(n,k). - Alois P. Heinz, Aug 19 2021