cp's OEIS Frontend

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

%I A282354 #15 Jun 20 2023 10:03:52
%S A282354 3,6,7,13,14,19,20,24,26,27,32,37,38,40,43,54,57,60,63,67,69,72,74,77,
%T A282354 79,84,85,86,87,88,97,103,108,109,111,114,115,125,126,127,132,133,134,
%U A282354 136,138,154,158,163,170,174,177,193,194,200,201,204,205,206,209
%N A282354 Positive j such that d(j) = d(j + 2*d(j)), where d(j) is the number of divisors of j.
%C A282354 The sequence contains the smaller member of every pair of cousin primes (A023200).
%C A282354 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.
%H A282354 Charles R Greathouse IV, <a href="/A282354/b282354.txt">Table of n, a(n) for n = 1..10000</a>
%t A282354 Select[Range@ 210, Function[d, DivisorSigma[0, # + 2 d] == d]@ DivisorSigma[0, #] &] (* _Michael De Vlieger_, Feb 13 2017 *)
%o A282354 (PARI) is(n)=my(d=numdiv(n)); d==numdiv(n+2*d) \\ _Charles R Greathouse IV_, Feb 14 2017
%Y A282354 Cf. A000005, A025487, A037916, A175304, A282231.
%K A282354 nonn
%O A282354 1,1
%A A282354 _Vladimir Shevelev_, Feb 13 2017
%E A282354 More terms from _Peter J. C. Moses_, Feb 13 2017