A161493 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.

1, 4, 9, 16, 64, 100, 144, 196, 225, 324, 441, 484, 576, 625, 676, 900, 1024, 1089, 1296, 1521, 1764, 1936, 2025, 2116, 2304, 2601, 3136, 3249, 3364, 3844, 4096, 4225, 4356, 4761, 4900, 5625, 5776, 6084, 6400, 6561, 6724, 7396, 7569, 8649, 8836, 9216, 9801

Offset

1,2

Comments

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

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]

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

Example

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.
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.

Mathematica

Select[Range[100]^2, OddQ@Quotient[#, DivisorSigma[0, #]] &] (* Ivan Neretin, Mar 23 2017 *)

Prog

(PARI) for(i=1, 999, k=i^2; if(floor(k/numdiv(k))%2, print1(k, ", "))) \\ Hagen von Eitzen, Jun 12 2009
(Python)
from sympy import divisor_count
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

Crossrefs

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

Sequence in context: A277699 A368891 A073723 * A030075 A296111 A038784

Adjacent sequences: A161490 A161491 A161492 * A161494 A161495 A161496

Keyword

nonn

Author

John W. Layman, Jun 11 2009

Status

approved