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A175304 A positive integer n is included if d(n+d(n)) = d(n), where d(n) is the number of divisors of n.

%I A175304 #55 Feb 08 2025 04:30:42
%S A175304 3,5,6,10,11,12,17,22,29,34,35,41,44,51,58,59,60,65,70,71,72,82,84,87,
%T A175304 91,92,96,101,102,107,111,115,118,119,125,128,129,130,137,141,142,147,
%U A175304 149,155,174,179,182,183,191,197,201,202,205,209,213,214,215,217,222
%N A175304 A positive integer n is included if d(n+d(n)) = d(n), where d(n) is the number of divisors of n.
%C A175304 The sequence contains the smaller member of every pair of twin primes (A001359) and all squarefree semiprimes m such that m+4 is also a squarefree semiprime (A255746). Can one prove that this is an infinite sequence? - _Vladimir Shevelev_, Jul 11 2015
%C A175304 The sequence does not contain perfect squares. Indeed, let a(m)=k^2. Then d(k^2+d(k^2)) = d(k^2). Note that d(k^2) is odd. On the other hand, it is known (A046522) that d(k^2)<2*k. Hence, (k+1)^2 - k^2 > d(k^2). Thus k^2<k^2+d(k^2)<(k+1)^2 and k^2+d(k^2) cannot be perfect squares. So, k^2+d(k^2) is even and we have a contradiction. - _Vladimir Shevelev_, Feb 10 2017
%C A175304 If p is prime and t+1 is odd prime, then p^t is not in the sequence. Indeed, if d(p^t+t+1)=t+1, then p^t+t+1=q^t, where q is prime > p (if p^t+t+1= say q^l*r^m, then (l+1)*(m+1)=t+1 which is impossible by the condition). But q>=p+2 and p^t+t+1>=p^t+2*t*p^(t-1) or t+1>=2*t*p^(t-1) which trivially has only solution t=1; however, by the condition t>=2. - _Vladimir Shevelev_, Feb 18 2017
%C A175304 If an odd integer k is in this sequence, so is 2k. - _Charlie Neder_, Jan 14 2019
%H A175304 Alois P. Heinz, <a href="/A175304/b175304.txt">Table of n, a(n) for n = 1..10000</a>
%e A175304 10 is in the sequence because d(10)=4 and d(10+d(10))=d(14)=4. - _Emeric Deutsch_, Apr 08 2010
%p A175304 with(numtheory): a := proc (n) if tau(n+tau(n)) = tau(n) then n else end if end proc: seq(a(n), n = 1 .. 230); # _Emeric Deutsch_, Apr 08 2010
%t A175304 Select[Range@ 224, Function[n, DivisorSigma[0, n + #] == # &@ DivisorSigma[0, n]]](* _Michael De Vlieger_, Sep 27 2015 *)
%t A175304 Position[#, 0][[All, 1]] &@ Table[DivisorSigma[0, n + DivisorSigma[0, n]] - DivisorSigma[0, n], {n, 222}] (* _Michael De Vlieger_, May 21 2017 *)
%o A175304 (PARI) is(n)=numdiv(n+n=numdiv(n))==n \\ _M. F. Hasler_, Sep 27 2015
%Y A175304 Cf. A000005, A062249, A001359, A255746, A259934, A282175, A282231, A286529.
%Y A175304 Positions of zeros in A286530.
%K A175304 nonn
%O A175304 1,1
%A A175304 _Leroy Quet_, Mar 24 2010
%E A175304 More terms from _Emeric Deutsch_, Apr 08 2010