A282354 Positive j such that d(j) = d(j + 2*d(j)), where d(j) is the number of divisors of j.

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

Offset

1,1

Comments

The sequence contains the smaller member of every pair of cousin primes (A023200).

The sequence contains no perfect squares. Indeed, let a(m) = k^2 for some m. Then, by the definition, d(k^2 + 2*d(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)^2 - k^2 = 4*k + 4 > 2*d(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 a 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: A293437 A359756 A350943 * 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