A394792 Number of positive integers <= n with exactly three odd divisors.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9

Offset

1,18

Comments

a(n) is also the number of positive integers <= n with 3 partitions into consecutive parts.

a(n) is also the number of positive integers m <= n whose symmetric representation of sigma(m) has 3 subparts.

For a theorem of correspondence between the partitions of n into k consecutive parts and the odd divisors of n see A379630.

Links

Table of n, a(n) for n=1..100.

Example

The first number with 3 odd divisors is 9 so a(9) = 1.
The second number with 3 odd divisors is 18 so a(18) = 2.
The third number with 3 odd divisors is 25 so a(25) = 3.
The three partitions of 9 into consecutive parts are [9], [5, 4], [4, 3, 2].
The three partitions of 18 into consecutive parts are [18], [7, 6, 5], [6, 5, 4, 3].
The three partitions of 25 into consecutive parts are [25], [13, 12], [7, 6, 5, 4, 3].

Mathematica

seq[n_] := Accumulate[Table[Boole[DivisorSigma[0, k/2^IntegerExponent[k, 2]] == 3], {k, 1, n}]]; seq[100] (* Amiram Eldar, Apr 04 2026 *)

Prog

(Python)
from math import isqrt
from sympy import primepi
def A394792(n): return 1-(m:=n.bit_length()-1)+sum(primepi(isqrt(n>>i)) for i in range(m)) # Chai Wah Wu, Apr 13 2026

Crossrefs

Column 3 of A394053.

Cf. A001227, A038547, A070939, A237593, A286000, A286001, A279387 (subparts), A299765, A379630, A394789.

Sequence in context: A372406 A329195 A204164 * A257639 A180447 A295866

Adjacent sequences: A394789 A394790 A394791 * A394793 A394794 A394795

Keyword

nonn,easy

Author

Omar E. Pol, Apr 04 2026

Status

approved