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Least k such that the product of n consecutive integers beginning with k exceeds n^n.
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%I #16 Mar 08 2023 09:40:35

%S 2,2,3,3,4,4,5,5,6,6,7,8,8,9,9,10,10,11,11,12,12,13,13,14,15,15,16,16,

%T 17,17,18,18,19,19,20,21,21,22,22,23,23,24,24,25,25,26,26,27,28,28,29,

%U 29,30,30,31,31,32,32,33,34,34,35,35,36,36,37,37,38,38,39,39,40,41,41

%N Least k such that the product of n consecutive integers beginning with k exceeds n^n.

%C Least k such that Pochhammer(k, n) > n^n. - _Ryan Propper_, Sep 06 2005

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.

%F Limit_{n->oo} a(n)/n = A090462. - _Ryan Propper_, Sep 06 2005

%e a(4) = 3: 2*3*4*5 = 120 < 256 = 4^4 < 3*4*5*6 = 360.

%t Do[k = 1; While[Pochhammer[k, n] <= n^n, k++ ]; Print[k], {n, 1, 100}] (* _Ryan Propper_, Sep 06 2005 *)

%o (PARI) b(n) = my(k=1); while(prod(i=k, k+n-1, i) <= n^n, k++); k; \\ _Michel Marcus_, Mar 08 2023

%Y Cf. A090462.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Aug 16 2005

%E More terms from _Ryan Propper_, Sep 06 2005