A272353 -id:A272353 - cp's OEIS frontend

A272872 Numbers k such that k+1 is divisible by number of divisors of k.

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 27, 29, 31, 35, 37, 39, 41, 43, 47, 51, 53, 55, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 123, 127, 131, 135, 137, 139, 143, 149, 151, 155, 157, 159, 163, 167

Offset: 1

Altug Alkan, May 11 2016

nonn

Inspired by A272353.
All odd primes are obvious members.
Numbers k such that k == -1 (mod A000005(k)). Nonprime terms are listed in A354714. - Max Alekseyev, Jun 04 2022
63 is the least number that is not in this sequence but is a member of A187929.

			15 is a term because A000005(15) = 4 divides 15+1 = 16.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Claudia A. Spiro, How Often Does the Number of Divisors of an Integer Divide Its Successor?, J. London Math. Soc. (1985) s2-31 (1): 30-40.

Cf. A000005, A187929, A272353, A033950, A354711, A354712, A354714, A354715, A354716.

Select[Range@167, Mod[#+1, DivisorSigma[0, #]] == 0 &] (* Giovanni Resta, May 21 2016 *)

lista(nn) = {for(n=1, nn, if((n+1) % numdiv(n) == 0, print1(n, ", ")));}

A268037 Numbers k such that the number of divisors of k+2 divides k and the number of divisors of k divides k+2.

Original entry on oeis.org

4, 30, 48, 110, 208, 270, 320, 368, 510, 590, 688, 750, 1070, 1216, 1328, 1566, 1808, 2030, 2190, 2510, 2670, 2768, 3008, 3088, 3728, 4110, 4208, 4430, 4528, 4688, 4698, 4910, 5008, 5696, 5870, 5886, 5968, 6128, 6592, 6846, 7088, 7310, 7790, 8384, 9008, 9230, 9390, 9488, 9534, 9710

Offset: 1

Waldemar Puszkarz, Apr 27 2016

nonn

One can call such pairs {n, n+2} mutually (or amicably) step 2 refactorable numbers.
All terms are even.
Proof. Let us suppose n is odd. Then so is n+2 and the number of their divisors has to be odd, which implies they are squares (see the proof in A268066). Since the separation between closest squares is 2n+1, it is always greater than 2, except for n=0, when it is 1.
Contains 48 + 160*k if 3 + 10*k and 5 + 16*k are prime. Dickson's conjecture implies that there are infinitely many of these. - Robert Israel, May 09 2016

			4 is a term because its number of divisors (3) divides 6=4+2 and the number of divisors of 6 (4) divides 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005 (number of divisors), A033950 (refactorable numbers), A272353, A269781 (related sequences).

select(n -> n mod numtheory:-tau(n+2)=0 and (n+2) mod numtheory:-tau(n) = 0,
2*[$1..10000]); # Robert Israel, May 09 2016

lst={}; Do[ If[ Divisible[n, DivisorSigma[0, n+2]]&&Divisible[n+2, DivisorSigma[0, n]], AppendTo[lst, n]], {n, 12000}]; lst
Select[Range[12000], Divisible[#, DivisorSigma[0, # + 2]] && Divisible[# + 2, DivisorSigma[0, #]] &]

for(n=1, 12000, (n%numdiv(n+2)==0)&&((n+2)%numdiv(n)==0)&&print1(n ", "))

from sympy import divisors
def ok(n): return n%len(divisors(n+2)) == 0 and (n+2)%len(divisors(n)) == 0
print(list(filter(ok, range(1, 9711)))) # Michael S. Branicky, Apr 30 2021

A269781 a(n) is the smallest k different from n such that (n, k) is an amicably refactorable pair (see the comments).

Original entry on oeis.org

4, 3, 16, 4, 64, 24, 36, 16, 1024, 6, 4096, 64, 4, 5, 65536, 12, 262144, 6, 4, 1024, 4194304, 8, 81, 4096, 4, 6, 268435456, 16, 1073741824, 6, 4, 65536, 16, 9, 68719476736, 262144, 4, 8, 1099511627776, 32, 4398046511104, 6, 36, 4194304, 70368744177664, 10, 729, 48, 4, 6, 4503599627370496, 32, 16, 8

Offset: 3

Waldemar Puszkarz, May 01 2016

nonn

Let m and k be distinct integers and numdiv(n) be the number of divisors of n (A000005(n)). We call m and k amicably refactorable if numdiv(m) divides k and numdiv(k) divides m.
For any n with no amicably refactorable partner, a(n) = 0.
Conjecture: the sequence contains no zeros.
1 does not have an amicable partner as all other numbers have more than one divisor and 2 does not have an amicable partner as all other numbers with two divisors are odd primes and cannot be divided by the number of divisors of 2, also 2. All other numbers may have an amicably refactorable partner, though for some, primes, semiprimes and squares of primes in particular, this number can be quite large.
For primes and semiprimes, a(n) = 2^(f(n) - 1), (see A061286), where f(n) is their largest prime factor. For squares of primes, a(n) = 3^(|sqrt(n)| - 1), except for n = 9 for which this formula yields 9; this forces us to choose the next best candidate: 36.

			For n=5, a(5)=16 as the number of divisors of n (2) divides a(n) while the number of divisors of a(n) (5) divides 5 and 16 is the smallest number for which this happens.

Cf. A000005 (number of divisors), A033950 (refactorable numbers), A061286 (subsequence for odd prime indices and semiprime indices), A268037, A272353 (related sequences).

A269781 = {}; Do[k = 1; If[PrimeQ[n] || PrimeNu[n] == 2 && PrimeOmega[n] == 2, AppendTo[A269781, 2^(First[Last[FactorInteger[n]]] - 1)], If[PrimeQ @ Sqrt @ n && (n > 9), AppendTo[A269781, 3^(Sqrt[n] - 1)],While[k != n && !(Divisible[n, DivisorSigma[0, k]] && Divisible[k, DivisorSigma[0, n]]), k++]; If[k == n, k = n + 1; While[!(Divisible[n, DivisorSigma[0, k]] && Divisible[k, DivisorSigma[0, n]]), k++]]; AppendTo[A269781, k]]], {n, 3, 56}]; A269781

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A272872 Numbers k such that k+1 is divisible by number of divisors of k.

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A268037 Numbers k such that the number of divisors of k+2 divides k and the number of divisors of k divides k+2.

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A269781 a(n) is the smallest k different from n such that (n, k) is an amicably refactorable pair (see the comments).

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