A267892 Numbers with 9 odd divisors.

225, 441, 450, 882, 900, 1089, 1225, 1521, 1764, 1800, 2178, 2450, 2601, 3025, 3042, 3249, 3528, 3600, 4225, 4356, 4761, 4900, 5202, 5929, 6050, 6084, 6498, 6561, 7056, 7200, 7225, 7569, 8281, 8450, 8649, 8712, 9025, 9522, 9800, 10404, 11858, 12100, 12168, 12321, 12996, 13122, 13225, 14112, 14161, 14400, 14450, 15129

Offset: 1

Author

Omar E. Pol, Apr 03 2016

Keywords

• nonn

Comments

Positive integers that have exactly nine odd divisors.

Numbers k such that the symmetric representation of sigma(k) has 9 subparts. - Omar E. Pol, Dec 29 2016

From Robert Israel, Dec 29 2016: (Start)

Numbers k such that A000265(k) is in A030627.

Numbers of the form 2^j*p^8 or 2^j*p^2*q^2 where p and q are distinct odd primes. (End)

Numbers that can be formed in exactly 8 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Column 9 of A266531.

Cf. A001227, A038547, A237593, A279387.

Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, A267891, this sequence, A267893.

Cf. A000265, A030627.

Programs

GAP

A:=List([1..16000],n->DivisorsInt(n));; B:=List([1..Length(A)],i->Filtered(A[i],IsOddInt));;
a:=Filtered([1..Length(B)],i->Length(B[i])=9); # Muniru A Asiru, Aug 14 2018

Maple

N:= 10^5: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..floor(sqrt(N)/2),2)]);
Aodd:= select(`<=`,map(t -> t^8, P) union {seq(seq(P[i]^2*P[j]^2,i=1..j-1),j=1..nops(P))}, N):
A:= map(t -> seq(2^j*t,j=0..ilog2(N/t)), Aodd):
sort(convert(A,list)); # Robert Israel, Dec 29 2016

Mathematica

Select[Range[5^6], Length[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] == 9 &] (* Michael De Vlieger, Apr 04 2016 *)
Select[Range[16000],Total[Boole[OddQ[Divisors[#]]]]==9&] (* Harvey P. Dale, May 12 2019 *)

PARI

isok(n) = sumdiv(n, d, (d%2)) == 9; \\ after Michel Marcus.

Formula

A001227(a(n)) = 9.

Sum_{n>=1} 1/a(n) = (P(2)-1/4)^2 - P(4) + 2*P(8) + 7/128 = 0.026721189882055998428..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 16 2024