OFFSET
1,1
COMMENTS
One may call such pairs {n, n+1} mutually (or amicably) one step refactorable numbers.
While most terms appear to be divisible by 3, some are not, the first being 2985983=7*11*13*19*157.
From Robert Israel, May 09 2016: (Start)
2^k-1 is a term if k+1 is an odd prime and 2^k-1 is squarefree.
If n is an odd term, then n+1 is a square, and n == 3 mod 4. (End)
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
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..2500
EXAMPLE
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.
MAPLE
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
MATHEMATICA
lst={}; Do[ If[ Divisible[n, DivisorSigma[0, n+1]]&&Divisible[n+1, DivisorSigma[0, n]], AppendTo[lst, n]], {n, 7000000}]; lst
Select[Range[7000000], Divisible[#, DivisorSigma[0, # + 1]] && Divisible[# + 1, DivisorSigma[0, #]] &]
PROG
(PARI) for(n=1, 7000000, (n%numdiv(n+1)==0) && ((n+1)%numdiv(n)==0)&& print1(n ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Waldemar Puszkarz, Apr 26 2016
STATUS
approved