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A345708 a(n) is the least positive number starting an interval of consecutive integers whose product of elements is n.

%I A345708 #25 Aug 15 2025 09:10:54
%S A345708 1,1,3,4,5,1,7,8,9,10,11,3,13,14,15,16,17,18,19,4,21,22,23,1,25,26,27,
%T A345708 28,29,5,31,32,33,34,35,36,37,38,39,40,41,6,43,44,45,46,47,48,49,50,
%U A345708 51,52,53,54,55,7,57,58,59,3,61,62,63,64,65,66,67,68,69
%N A345708 a(n) is the least positive number starting an interval of consecutive integers whose product of elements is n.
%C A345708 This sequence is similar to A118235; here we multiply, there we add.
%C A345708 a(n) is the index of the first row of A068424 (interpreted as a square array) containing n.
%C A345708 If n is the product of k consecutive integers, then k! divides n.
%H A345708 Rémy Sigrist, <a href="/A345708/b345708.txt">Table of n, a(n) for n = 1..10000</a>
%F A345708 a(n) = 1 iff n is a factorial number (A000142).
%F A345708 a(n) <> 2.
%F A345708 a(n) = 3 iff n >= 3 and n belongs to A001710.
%F A345708 a(n) <= n.
%F A345708 a(p! / (n-1)!) = n for any n >= 3 and any prime number p >= n.
%F A345708 a(q) = q for any prime power q > 2.
%F A345708 a(n) = n for any odd number n.
%F A345708 a(n) < n iff n belongs to A045619.
%e A345708 The square array A068424(n, k) begins:
%e A345708   n\k|   1    2     3      4       5        6
%e A345708   ---+---------------------------------------
%e A345708     1|   1    2     6     24     120      720
%e A345708     2|   2    6    24    120     720     5040
%e A345708     3|   3   12    60    360    2520    20160
%e A345708     4|   4   20   120    840    6720    60480
%e A345708 - so a(1) = a(2) = a(6) = a(24) = a(120) = a(720) = 1,
%e A345708      a(3) = a(12) = a(60) = a(360) = 3,
%e A345708      a(4) = a(20) = 4.
%o A345708 (PARI) a(n) = { fordiv (n, d, my (r=n); for (k=d, oo, if (r==1, return (d), r%k, break, r/=k))) }
%o A345708 (PARI) a(n) = { for (i=2, oo, if (n%i!, forstep (j=i-1, 2, -1, my (d=sqrtnint(n,j)); while (d-1 && n%(d-1)==0, d--); while (n%d==0, my (r=n); for
%o A345708 (k=d, oo, if (r==1, return (if (d==2, 1, d)), r%k, break, r/=k)); d++)); break)); return (n) }
%o A345708 (Python)
%o A345708 from sympy import divisors
%o A345708 def a(n):
%o A345708     if n%2 == 0: return n
%o A345708     divs = divisors(n)
%o A345708     for i, d in enumerate(divs[:len(divs)//2]):
%o A345708         p = lastj = d
%o A345708         for j in divs[i+1:]:
%o A345708             if p >= n or j - lastj > 1: break
%o A345708             p, lastj = p*j, j
%o A345708         if p == n: return d
%o A345708     return n
%o A345708 print([a(n) for n in range(1, 70)]) # _Michael S. Branicky_, Jun 29 2021
%Y A345708 Cf. A000142, A001710, A045619, A068424, A118235, A246655.
%K A345708 nonn
%O A345708 1,3
%A A345708 _Rémy Sigrist_, Jun 24 2021