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

Author

Vladimir Shevelev, Feb 13 2017

Keywords

• nonn

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

Crossrefs

Cf. A000005, A025487, A037916, A175304, A282231.

Programs

Mathematica

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

PARI

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

Extensions

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