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%I #26 Mar 17 2024 03:12:24
%S 1,1,1,1,1,2,1,1,2,3,1,2,2,4,1,2,3,2,4,4,5,3,2,4,6,4,8,3,5,1,9,6,4,8,
%T 12,5,3,7,9,6,10,2,18,12,8,16,9,15,3,7,6,14,18,12,20,4,36,15,9,16,5,
%U 27,18,30,6,14,12,28,36,24,25,15,8,27,45,9,21,18
%N First differences of 5-smooth numbers (A051037).
%C lim inf a(n) >= 2 by Størmer's theorem. Is lim a(n) = infinity? It would be very surprising if this were false, since then there is some k such that n and n+k are both 5-smooth for infinitely many n. - _Charles R Greathouse IV_, Nov 28 2012
%C A085152 gives all n's for which a(n) = 1. Thue-Siegel-Roth theorem gives lim a(n) = infinity. With the aid of lower bounds for linear forms in logarithms, Tijdeman showed that a(n+1)-a(n) > a(n)/(log a(n))^C for some effectively computable constant C. - _Tomohiro Yamada_, Apr 15 2017
%H Michael De Vlieger, <a href="/A219794/b219794.txt">Table of n, a(n) for n = 1..10000</a> (First 1000 terms from Zak Seidov)
%H Carl Størmer, <a href="https://archive.org/stream/skrifterudgivnea1897chri#page/n81/mode/2up">Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications</a>, Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.
%H Robert Tijdeman, <a href="http://www.numdam.org/item?id=CM_1973__26_3_319_0">On integers with many small prime factors</a>, Compos. Math. 26 (1973), 319-330.
%F a(n) = A051037(n+1) - A051037(n).
%t Differences@ Union@ Flatten@ Table[2^i * 3^j * 5^k, {i, 0, Log2[#]}, {j, 0, Log[3, #/(2^i)]}, {k, 0, Log[5, #/(2^i*3^j)] } ] &[1000] (* _Michael De Vlieger_, Mar 16 2024 *)
%Y Cf. A051037, A085152.
%K nonn
%O 1,6
%A _Zak Seidov_, Nov 28 2012
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