cp's OEIS Frontend

If k is in the sequence then so is k*m for positive m. - David A. Corneth, Aug 19 2022

Numbers that are divisible by at least one of 7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138. For proof, see link. - Robert Israel, Sep 02 2022

The asymptotic density of this sequence is 17819629/37182145 = 0.479252... . - Amiram Eldar, Aug 08 2023

Terms are even. Proof by contradiction. Suppose m = a(n) is odd. Then each divisor is odd. Adding 8 odd numbers gives an even number. A contradiction. - David A. Corneth, Sep 02 2022

If k is in the sequence then so is k*m. - David A. Corneth, Oct 08 2022

From David A. Corneth, Oct 08 2022: (Start)

All terms are even. Proof: suppose a term is odd. Then all divisors are odd. Adding 10 odd numbers gives an even number. A contradiction.

If k is a term then so is k*m for m >= 1. Proof: Multiply each divisor in this sum of 10 divisors that give k with m. Then each term is a divisor of k*m and their sum is k*m. (End)

Slabs on the sidewalk are numbered n = 0, 1, 2,... and each has a label L(n) = 1, 1, 2, 2, 3, 3,...

At a given slab, a jump can be taken forward or backward by L(n) places (but not back before slab 0).

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For every n >= 0,

let L(n) = 1 + floor(n/2) -- the label on slab n,

let forward(n) = n + L(n) -- jumping forward,

if n > 0, let backward(n) = n - L(n) -- jumping backward,

let lambda(n) = floor((2/3)*n),

let mu(n) = 1 + 2*n,

let nu(n) = 2 + 2*n.

Observe that given n >= 0, there are exactly two ways of landing onto slab n with a direct jump backwards:

backward-jumping from mu(n) to n, and

backward-jumping from nu(n) to n.

If n is a multiple of 3, there is no other ways of jumping onto slab n. But if n is not a multiple of 3, there is one additional way:

forward-jumping from lambda(n) to n.

(Note that L is A008619, forward(n) == A006999(n+1), lambda is A004523, mu is A005408, nu is A299174.)

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Every slab n > 0 is reachable from slab 0, since there always exists some slab s < n which reaches n by one or more jumps:

if n != 0 (mod 3), then s = lambda(n) = floor((2/3)*n) takes one forward jump to n,

if n == 0 (mod 3) but n != 0 (mod 9), then s = lambda o lambda o mu(n) = floor((8/9)*n) takes two forward jumps and one backward jump to n,

if n == 0 (mod 9), then s = lambda o lambda o nu(n) = floor((8/9)*n + 6/9) takes two forward jumps and one backward jump to n.

This demonstrates that the sequence never stops.

This also gives the following bounds:

a(n) <= 1 + (4/3)*n,

a(n) <= 6*log(n)/log(9/8).

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The sequence is a surjective mapping N -> N, since given any n >= 0:

a(forward^n(0)) == n.

Numbers which can be expressed as a sum of a set of positive consecutive even numbers: sum_{i=m..m+k} A005843(i), m>=1, k>=1.

Differs from A054741, which contains 105 for example.

These are the numbers that are not free of odd prime factors, but are not odd. Compare with A051144, nonsquarefree nonsquares. The self-inverse function defined by A225546 maps the members of either set 1:1 onto the other set. - Peter Munn, Jul 31 2020 with edit Feb 14 2022

Equivalently, numbers k such that totient(k) <= cototient(k).

Using the primes up to 23 it is possible to show that this sequence has (lower) density greater than 0.51. - Charles R Greathouse IV, Oct 26 2015

The asymptotic density of this sequence is in the interval (0.51120, 0.51176) (Kobayashi, 2016, improving the bounds 0.5105 and 0.5241 that were given by Wall, 1972). - Amiram Eldar, Oct 15 2020

A permutation of A028983.

A007417 (which has asymptotic density 3/4) lists index n such that a(n) = 3n. The sequence maps the terms of A007417 1:1 onto A145204\{0}, defining a bijection between them.

Similarly, bijections are defined from the odd numbers (A005408) to the nonsquare odd numbers (A088828), from the positive even numbers (A299174) to A088829, from A003159 to the nonsquares in A003159, and from A325424 to the nonsquares in A036668. The latter two bijections are between sets where membership depends on whether a number's squarefree part divides by 2 and/or 3.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.