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A267892 Numbers with 9 odd divisors.

%I A267892 #39 Sep 16 2024 02:10:26
%S A267892 225,441,450,882,900,1089,1225,1521,1764,1800,2178,2450,2601,3025,
%T A267892 3042,3249,3528,3600,4225,4356,4761,4900,5202,5929,6050,6084,6498,
%U A267892 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
%N A267892 Numbers with 9 odd divisors.
%C A267892 Positive integers that have exactly nine odd divisors.
%C A267892 Numbers k such that the symmetric representation of sigma(k) has 9 subparts. - _Omar E. Pol_, Dec 29 2016
%C A267892 From _Robert Israel_, Dec 29 2016: (Start)
%C A267892 Numbers k such that A000265(k) is in A030627.
%C A267892 Numbers of the form 2^j*p^8 or 2^j*p^2*q^2 where p and q are distinct odd primes. (End)
%C A267892 Numbers that can be formed in exactly 8 ways by summing sequences of 2 or more consecutive positive integers. - _Julie Jones_, Aug 13 2018
%H A267892 Robert Israel, <a href="/A267892/b267892.txt">Table of n, a(n) for n = 1..10000</a>
%F A267892 A001227(a(n)) = 9.
%F A267892 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
%p A267892 N:= 10^5: # to get all terms <= N
%p A267892 P:= select(isprime, [seq(i,i=3..floor(sqrt(N)/2),2)]);
%p A267892 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):
%p A267892 A:= map(t -> seq(2^j*t,j=0..ilog2(N/t)), Aodd):
%p A267892 sort(convert(A,list)); # _Robert Israel_, Dec 29 2016
%t A267892 Select[Range[5^6], Length[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] == 9 &] (* _Michael De Vlieger_, Apr 04 2016 *)
%t A267892 Select[Range[16000],Total[Boole[OddQ[Divisors[#]]]]==9&] (* _Harvey P. Dale_, May 12 2019 *)
%o A267892 (PARI) isok(n) = sumdiv(n, d, (d%2)) == 9; \\ after _Michel Marcus_.
%o A267892 (GAP) A:=List([1..16000],n->DivisorsInt(n));; B:=List([1..Length(A)],i->Filtered(A[i],IsOddInt));;
%o A267892 a:=Filtered([1..Length(B)],i->Length(B[i])=9); # _Muniru A Asiru_, Aug 14 2018
%Y A267892 Column 9 of A266531.
%Y A267892 Cf. A001227, A038547, A237593, A279387.
%Y A267892 Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, A267891, this sequence, A267893.
%Y A267892 Cf. A000265, A030627.
%K A267892 nonn
%O A267892 1,1
%A A267892 _Omar E. Pol_, Apr 03 2016