A394789 Number of positive integers <= n with exactly two odd divisors.

0, 0, 1, 1, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 9, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 29, 29, 30, 31, 31, 31, 32, 32, 33, 34, 34, 35, 36, 36, 36, 36, 36, 37, 38, 38, 38, 39, 39, 40, 41, 41, 42, 42, 42

Offset

1,5

Comments

a(n) is also the number of positive integers <= n with two partitions into consecutive parts.

a(n) is also the number of positive integers m <= n whose symmetric representation of sigma(m) has two subparts.

Links

Table of n, a(n) for n=1..78.

Example

The first number with two odd divisors is 3 so a(3) = 1.
4 has only one odd divisor so a(4) = a(3) = 1.
The second number with two odd divisors is 5 so a(5) = 2.
The third number with two odd divisors is 6 so a(6) = 3.
The two partitions of 3 into consecutive parts are [3], [2, 1].
The two partitions of 5 into consecutive parts are [5], [3, 2].
The two partitions of 6 into consecutive parts are [6], [3, 2, 1].

Mathematica

seq[n_] := Accumulate[Table[Boole[DivisorSigma[0, k/2^IntegerExponent[k, 2]] == 2], {k, 1, n}]]; seq[100] (* Amiram Eldar, Apr 03 2026 *)

Prog

(Python)
from sympy import primepi
def A394789(n): return -(m:=n.bit_length()-1)+sum(primepi(n>>i) for i in range(m)) # Chai Wah Wu, Apr 13 2026

Crossrefs

Column 2 of A394053.

Cf. A001227, A038547, A070939, A237593, A286000, A286001, A279387 (subparts), A299765.

Sequence in context: A099479 A120508 A199332 * A029085 A087875 A195848

Adjacent sequences: A394786 A394787 A394788 * A394790 A394791 A394792

Keyword

nonn,easy

Author

Omar E. Pol, Apr 03 2026

Status

approved