

A282354


Positive n such that d(n) = d(n+2*d(n)), where d(n) is the number of divisors of n.


2



3, 6, 7, 13, 14, 19, 20, 24, 26, 27, 32, 37, 38, 40, 43, 54, 57, 60, 63, 67, 69, 72, 74, 77, 79, 84, 85, 86, 87, 88, 97, 103, 108, 109, 111, 114, 115, 125, 126, 127, 132, 133, 134, 136, 138, 154, 158, 163, 170, 174, 177, 193, 194, 200, 201, 204, 205, 206, 209
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OFFSET

1,1


COMMENTS

The sequence contains the smaller member of every pair of cousin primes (A023200).
The sequence does not contain perfect squares. Indeed, let for some m a(m)=k^2. Then, by the definition, d(k^2+2d(k^2))=d(k^2). Note that d(k^2) is odd. On the other hand, it is known (cf. A046522) that d(k^2)<2*k. Hence (k+2)^2k^2=4*k+1>2d(k^2). Thus k^2< k^2+2*d(k^2 <(k+2)^2. Since, evidently, k^2+2*d(k^2) cannot be (k+1)^2, then k^2 + 2*d(k^2) cannot be a square. Therefore, d(k^2+2*d(k^2)) is even, which is contradiction.


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000


MATHEMATICA

Select[Range@ 210, Function[d, DivisorSigma[0, # + 2 d] == d]@ DivisorSigma[0, #] &] (* Michael De Vlieger, Feb 13 2017 *)


PROG

(PARI) is(n)=my(d=numdiv(n)); d==numdiv(n+2*d) \\ Charles R Greathouse IV, Feb 14 2017


CROSSREFS

Cf. A000005, A025487, A037916, A175304, A282231.
Sequence in context: A294231 A280873 A293437 * A088146 A176301 A191290
Adjacent sequences: A282351 A282352 A282353 * A282355 A282356 A282357


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Feb 13 2017


EXTENSIONS

More terms from Peter J. C. Moses, Feb 13 2017


STATUS

approved



