cp's OEIS Frontend

A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

%I A069562 #40 Jun 06 2025 09:34:30
%S A069562 9,18,25,36,49,50,72,81,98,100,121,144,162,169,196,200,225,242,288,
%T A069562 289,324,338,361,392,400,441,450,484,529,576,578,625,648,676,722,729,
%U A069562 784,800,841,882,900,961,968,1058,1089,1152,1156,1225,1250,1296,1352,1369
%N A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.
%C A069562 Previous name: sum(d|n,6d/(2+mu(d))) is odd, where mu(.) is the Moebius function, A008683.
%C A069562 From _Peter Munn_, Jul 06 2020: (Start)
%C A069562 Numbers that have an odd number of odd nonsquarefree divisors.
%C A069562 [Proof of equivalence to the name, where m denotes a positive integer:
%C A069562 (1) These properties are equivalent: (a) m has an even number of odd squarefree divisors; (b) m has a nontrivial odd part.
%C A069562 (2) These properties are equivalent: (a) m has an odd number of odd divisors; (b) the odd part of m is square.
%C A069562 (3) m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are both true or both false.
%C A069562 (4) The trivial odd part, 1, is a square, so (1)(b) and (2)(b) cannot both be false, which (from (1), (2)) means (1)(a) and (2)(a) cannot both be false.
%C A069562 (5) From (3), (4), m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are true.
%C A069562 (6) m satisfies the condition in the name if and only if (1)(b) and (2)(b) are true, which (from (1), (2)) is equivalent to (1)(a) and (2)(a) being true, and hence from (5), to m satisfying the condition at the start of this comment.]
%C A069562 (End)
%C A069562 Numbers whose sum of non-unitary divisors (A048146) is odd. - _Amiram Eldar_, Sep 16 2024
%H A069562 David A. Corneth, <a href="/A069562/b069562.txt">Table of n, a(n) for n = 1..10000</a>
%H A069562 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddPart.html">Odd part</a>.
%F A069562 Sum_{n>=1} 1/a(n) = 2 * Sum_{k>=1} 1/(2*k+1)^2 = Pi^2/4 - 2 = A091476 - 2 = 0.467401... - _Amiram Eldar_, Feb 18 2021
%e A069562 To determine the odd part of 18, remove all factors of 2, leaving 9. 9 is a nontrivial square, so 18 is in the sequence. - _Peter Munn_, Jul 06 2020
%t A069562 Select[Range[1000], (odd = #/2^IntegerExponent[#, 2]) > 1 && IntegerQ @ Sqrt[odd] &] (* _Amiram Eldar_, Sep 29 2020 *)
%o A069562 (PARI) upto(n) = { my(res = List()); forstep(i = 3, sqrtint(n), 2, for(j = 0, logint(n\i^2, 2), listput(res, i^2<<j) ) ); listsort(res); res } \\ _David A. Corneth_, Sep 28 2020
%Y A069562 A000265, A008683 are used in definitions of this sequence.
%Y A069562 Lists of numbers whose odd part satisfies other conditions: A028982 (square), A028983 (nonsquare), A029747 (less than 6), A029750 (less than 8), A036349 (even number of prime factors), A038550 (prime), A070776 U {1} (power of a prime), A072502 (square of a prime), A091067 (has form 4k+3), A091072 (has form 4k+1), A093641 (noncomposite), A105441 (composite), A116451 (greater than 4), A116882 (less than or equal to even part), A116883 (greater than or equal to even part), A122132 (squarefree), A229829 (7-rough), A236206 (11-rough), A260488\{0} (has form 6k+1), A325359 (proper prime power), A335657 (odd number of prime factors), A336101 (prime power).
%Y A069562 Cf. A048146, A091476.
%K A069562 easy,nonn
%O A069562 1,1
%A A069562 _Benoit Cloitre_, Apr 18 2002
%E A069562 New name from _Peter Munn_, Jul 06 2020