OFFSET
1,2
COMMENTS
The graph of the sequence shows similarities to the EKG sequence A064413, except that here it is the prime powers, in particular the powers of two, that cause the large blips in the graph as opposed to the primes themselves. In the terms studied, all terms a(n) = 2^k for k>=4 are preceded by 3*2^k and followed by 5*2^k, the later term being approximately at index n = 12*2^(k-2). It is likely this behavior is true for all k.
Terms that contain only prime exponents greater than 1 in their prime factorization also cause small blips in the graph. The vast majority of other terms are either equal to n itself, hence are fixed points, or are just 1 or 2 more than n.
The sequence is conjectured to be a permutation of the positive integers.
LINKS
Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Line graph of the first 500 terms.
EXAMPLE
a(5) = 6 = 2^1*3^1 as a(4) = 5 = 5^1, 6 is unused, and both 5 and 6 have the prime exponent 1 in their prime factorizations.
a(10) = 4 = 2^2 as a(9) = 12 = 2^2*3^1, 4 is unused, and both 4 and 12 have the prime exponent 2 in their prime factorizations.
MATHEMATICA
s={1, 2}; Do[i=2; Until[!MemberQ[s, i]&&ContainsAny[Last/@FactorInteger[s[[-1]]], Last/@FactorInteger[i]], i++]; AppendTo[s, i], {n, 64}]; s (* James C. McMahon, Sep 21 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Sep 19 2025
STATUS
approved