A161493 - OEIS

%I #30 May 09 2021 04:55:46

%S 1,4,9,16,64,100,144,196,225,324,441,484,576,625,676,900,1024,1089,

%T 1296,1521,1764,1936,2025,2116,2304,2601,3136,3249,3364,3844,4096,

%U 4225,4356,4761,4900,5625,5776,6084,6400,6561,6724,7396,7569,8649,8836,9216,9801

%N Positive integers, k, for which k mod d(k) and k have opposite (odd/even) parity, where d(k) is the number of divisors of k.

%C It appears that the sequence {a(n)} consists entirely of squares. (This has been verified to a(431) = 998001 = 999^2.)

%C A number k appears in the sequence if and only if k is a square and floor(k/d(k)) is odd. This is because k mod d(k) = k - d(k) * floor(k/d(k)) and d(k) is odd if and only if k is square. [_Hagen von Eitzen_, Jun 12 2009]

%H Ivan Neretin, <a href="/A161493/b161493.txt">Table of n, a(n) for n = 1..10000</a>

%e k=4 has three divisors, so 4 mod d(4) = 1, which is odd. But 4 is even. Therefore 4 is a term of the sequence.

%e k=25 has three divisors, so 25 mod d(25) = 1, which is odd. 25 is also odd. Therefore 25 is not a term of the sequence.

%t Select[Range[100]^2, OddQ@Quotient[#, DivisorSigma[0, #]] &] (* _Ivan Neretin_, Mar 23 2017 *)

%o (PARI) for(i=1,999,k=i^2;if(floor(k/numdiv(k))%2,print1(k,","))) \\ _Hagen von Eitzen_, Jun 12 2009

%o (Python)

%o from sympy import divisor_count

%o print([n**2 for n in range(1, 1001) if int(math.floor(n**2/divisor_count(n**2)))%2]) # _Indranil Ghosh_, Mar 23 2017

%Y Cf. A000005, A161494 (gives the square roots).

%K nonn

%O 1,2

%A _John W. Layman_, Jun 11 2009