OFFSET
1,2
COMMENTS
Every nonprime appears exactly once in this sequence.
If n is a square we can take t=1 and a(n) = n. If n is a prime > 3, then a(n) = 2n and t=3. If n is twice a prime, say p, then a(n) = 3p most of the time. The sequence b_1 < b_2 < ... < b_t will not contain either perfect squares or primes for they bring nothing to the solution. Also I know of no n such that t = 2. - Robert G. Wilson v, Jan 30 2002
Let k be a fixed integer and p be a prime, then a(k*p) = (k+1)*p for sufficiently large p. - Peter Kagey, Feb 03 2015
From David A. Corneth, Oct 26 2016: (Start)
Is for all k*p in A277624, a(k*p) = (k+1) * p?
Conjecture: Let b(n) = A006530(A007913(n)). If b(n)^2 >= 2 * n then a(n) = n + b(n) except for n = 3, 10, and 171.
(End)
a(n) <= A072905(n).
a(n) <= 2*n for all n > 3.
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., Problem 4.39, pages 147, 616, 533. [Reference revised by N. J. A. Sloane, Jan 13 2014]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Peter Kagey, Table of n, a(n) for n = 1..10000
R. L. Graham, Bijection between integers and composites, Problem 1242, Math. Mag., 60 (1987), p. 180. [Note that unless you subscribe to JSTOR this link will only show page 178, which contains a different problem proposed by R. L. Graham. - N. J. A. Sloane, Jan 13 2014]
Peter Kagey and Krishna Rajesh, On a Conjecture about Ron Graham's Sequence, arXiv:2410.04728 [math.NT], 2024.
FORMULA
If n is a square we can take t=1 and a(n)=n.
a(n) = A092487(n) + n. - Peter Kagey, Oct 22 2016
EXAMPLE
a(2) = 6 because the best such sequence is 2,3,6.
For n = 3 through 6 the {smallest m then smallest t then smallest product} solutions are 3,6,8; 4; 5,8,10; 6,8,12.
MATHEMATICA
Table[k = 0; Which[IntegerQ@ Sqrt@ n, k, And[PrimeQ@ n, n > 3], k = n, True, While[Length@ Select[n Map[Times @@ # &, n + Rest@ Subsets@ Range@ k], IntegerQ@ Sqrt@ # &] == 0, k++]]; k + n, {n, 40}] (* Michael De Vlieger, Oct 26 2016 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Jan 30 2002
Erroneous program (pointed out by Peter Kagey) removed by Reinhard Zumkeller, Nov 28 2014
STATUS
approved