cp's OEIS Frontend

A272353 Numbers n such that the number of divisors of n+1 divides n and the number of divisors of n divides n+1.

%I A272353 #33 May 18 2016 19:26:29
%S A272353 3,15,1023,6399,10815,15375,26895,53823,55695,65535,80655,107583,
%T A272353 118335,262143,309135,440895,614655,633615,817215,891135,1236543,
%U A272353 1784895,2676495,2715903,2849343,2985983,3182655,3225615,3268863,4194303,4326399,4343055,4596735,5053503
%N A272353 Numbers n such that the number of divisors of n+1 divides n and the number of divisors of n divides n+1.
%C A272353 One may call such pairs {n, n+1} mutually (or amicably) one step refactorable numbers.
%C A272353 While most terms appear to be divisible by 3, some are not, the first being 2985983=7*11*13*19*157.
%C A272353 From _Robert Israel_, May 09 2016: (Start)
%C A272353 2^k-1 is a term if k+1 is an odd prime and 2^k-1 is squarefree.
%C A272353 If n is an odd term, then n+1 is a square, and n == 3 mod 4. (End)
%C A272353 There is no term such that last digit of it 1 or 7. Proof: If n is an odd term, then n+1 is a square. For any odd number k, last digit can be trivially 1, 3, 5, 7, 9, that is, the last digit of k+1 is 2, 4, 6, 8, 0 for corresponding odd k values. There cannot be square such that last digit of it 2 or 8. So in this sequence, there is no term such that the last digit of it 1 or 7. - _Altug Alkan_, May 11 2016
%H A272353 Giovanni Resta, <a href="/A272353/b272353.txt">Table of n, a(n) for n = 1..2500</a>
%e A272353 15 is a term because the number of divisors of 16=15+1, which is 5, divides 15, and the number of divisors of 15, which is 4, divides 16.
%p A272353 select(t -> (t+1) mod numtheory:-tau(t) = 0 and t mod numtheory:-tau(t+1) = 0, [$1..10^6]); # _Robert Israel_, May 09 2016
%t A272353 lst={}; Do[ If[ Divisible[n, DivisorSigma[0, n+1]]&&Divisible[n+1, DivisorSigma[0, n]], AppendTo[lst, n]], {n, 7000000}]; lst
%t A272353 Select[Range[7000000], Divisible[#, DivisorSigma[0, # + 1]] && Divisible[# + 1, DivisorSigma[0, #]] &]
%o A272353 (PARI) for(n=1, 7000000, (n%numdiv(n+1)==0) && ((n+1)%numdiv(n)==0)&& print1(n ", "))
%Y A272353 Cf. A000005 (number of divisors), A033950 (refactorable numbers), A268037, A269781 (related sequences).
%K A272353 nonn
%O A272353 1,1
%A A272353 _Waldemar Puszkarz_, Apr 26 2016