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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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 xrange(1, 10001) if int(math.floor(n**2/divisor_count(n**2)))%2 == 1] # Indranil Ghosh, Mar 23 2017

CROSSREFS

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

Sequence in context: A023110 A277699 A073723 * A030075 A296111 A038784

Adjacent sequences:  A161490 A161491 A161492 * A161494 A161495 A161496

KEYWORD

nonn

AUTHOR

John W. Layman, Jun 11 2009

STATUS

approved

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Last modified January 22 09:52 EST 2019. Contains 319363 sequences. (Running on oeis4.)