OFFSET
1,2
COMMENTS
Not all (nonempty) subsets need to be considered. For example if some number m is the only number in {k, k+1, ..., k+n-1} that is divisible by p then m cannot be a factor of a number that occurs as a product of two subsets. - David A. Corneth, Feb 06 2025
Equivalently, for n >= 2, a(n) is the smallest k > 0 such that there is no nontrivial solution (x(k), ..., x(k+n-1)) to the system of linear equations e(k,p)*x(k) + ... + e(k+n-1,p)*x(k+n-1) = 0 (one equation for each prime p that appears as a factor of some number m in the range k <= m <= k+n-1), with all x(i) in {-1,0,1}. Here, e(m,p) is the exponent of the prime p in m: m = 2^e(m,2)*3^e(m,3)*5^e(m,5)*... . The system of equations can be reduced by using the ideas in the preceding comment. A solution x corresponds to the two sets of numbers m with x(m) = 1 and x(m) = -1, respectively, having the same product. - Pontus von Brömssen, Feb 08 2025
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..271 (terms 1..100 from David A. Corneth)
Michael S. Branicky, Python program
FORMULA
a(n) >= A239276(n).
From Michael S. Branicky, Feb 04 2025: (Start)
a(n+1) >= a(n).
a(n+1) = a(n) if a(n)+n is prime, is coprime to a(n)..a(n)+n-1, or has at least one prime factor that is coprime to a(n)..a(n)+n-1, for n > 1. (End)
For n >= 2, a(n) is the least k such that A380998(k) >= n. The reason that this does not hold for n = 1 is that the empty subset is permitted in A380998. - Pontus von Brömssen, Feb 12 2025
EXAMPLE
a(5) = 3 because the range {1,...,5} is ruled out by 1*2=2, the range {2,...,6} by 2*3 = 6 and all 31 subsets of {3,...,7} give a distinct product.
MATHEMATICA
a[1]=1; a[n_] := a[n] = Block[{k = a[n-1]}, While[Min@ Differences@ Sort[Times @@@ Rest@ Subsets@ Range[k, n+k-1]] == 0, k++]; k]; Array[a, 16]
PROG
(Python) # see link for a faster version
from itertools import count
def a(n, startfrom=2):
if n == 1: return 1
for k in count(max(2, startfrom)):
P, passes = set(), True
for i in range(k, k+n):
for p in list(P):
if i*p in P:
passes = False
break
P.add(i*p)
if not passes or i in P:
break
P.add(i)
if passes:
return k
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Feb 04 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Giovanni Resta, Mar 14 2014
EXTENSIONS
a(25)-a(27) from Pontus von Brömssen, Feb 04 2025
a(28)-a(29) from Michael S. Branicky, Feb 04 2025
a(30)-a(31) from Michael S. Branicky, Feb 05 2025
More terms from David A. Corneth, Feb 06 2025
STATUS
approved
