A272353 - OEIS

%I #33 May 18 2016 19:26:29

%S 3,15,1023,6399,10815,15375,26895,53823,55695,65535,80655,107583,

%T 118335,262143,309135,440895,614655,633615,817215,891135,1236543,

%U 1784895,2676495,2715903,2849343,2985983,3182655,3225615,3268863,4194303,4326399,4343055,4596735,5053503

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

%C One may call such pairs {n, n+1} mutually (or amicably) one step refactorable numbers.

%C While most terms appear to be divisible by 3, some are not, the first being 2985983=7*11*13*19*157.

%C From _Robert Israel_, May 09 2016: (Start)

%C 2^k-1 is a term if k+1 is an odd prime and 2^k-1 is squarefree.

%C If n is an odd term, then n+1 is a square, and n == 3 mod 4. (End)

%C 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 Giovanni Resta, <a href="/A272353/b272353.txt">Table of n, a(n) for n = 1..2500</a>

%e 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 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 lst={}; Do[ If[ Divisible[n, DivisorSigma[0, n+1]]&&Divisible[n+1, DivisorSigma[0, n]], AppendTo[lst, n]], {n, 7000000}]; lst

%t Select[Range[7000000], Divisible[#, DivisorSigma[0, # + 1]] && Divisible[# + 1, DivisorSigma[0, #]] &]

%o (PARI) for(n=1, 7000000, (n%numdiv(n+1)==0) && ((n+1)%numdiv(n)==0)&& print1(n ", "))

%Y Cf. A000005 (number of divisors), A033950 (refactorable numbers), A268037, A269781 (related sequences).

%K nonn

%O 1,1

%A _Waldemar Puszkarz_, Apr 26 2016