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

9, 18, 25, 36, 49, 50, 72, 81, 98, 100, 121, 144, 162, 169, 196, 200, 225, 242, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1058, 1089, 1152, 1156, 1225, 1250, 1296, 1352, 1369

Offset: 1

Author

Benoit Cloitre, Apr 18 2002

Keywords

• easy
• nonn

Comments

Previous name: sum(d|n,6d/(2+mu(d))) is odd, where mu(.) is the Moebius function, A008683.

From Peter Munn, Jul 06 2020: (Start)

Numbers that have an odd number of odd nonsquarefree divisors.

[Proof of equivalence to the name, where m denotes a positive integer:

(1) These properties are equivalent: (a) m has an even number of odd squarefree divisors; (b) m has a nontrivial odd part.

(2) These properties are equivalent: (a) m has an odd number of odd divisors; (b) the odd part of m is square.

(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.

(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.

(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.

(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.]

(End)

Numbers whose sum of non-unitary divisors (A048146) is odd. - Amiram Eldar, Sep 16 2024

Examples

			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
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Odd part.

Crossrefs

A000265, A008683 are used in definitions of this sequence.

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).

Cf. A048146, A091476.

Programs

Mathematica

Select[Range[1000], (odd = #/2^IntegerExponent[#, 2]) > 1 && IntegerQ @ Sqrt[odd] &] (* Amiram Eldar, Sep 29 2020 *)

PARI

upto(n) = { my(res = List()); forstep(i = 3, sqrtint(n), 2, for(j = 0, logint(n\i^2, 2), listput(res, i^2<David A. Corneth, Sep 28 2020

Formula

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

Extensions

New name from Peter Munn, Jul 06 2020