A219794 First differences of 5-smooth numbers (A051037).

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, 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, 27, 18, 30, 6, 14, 12, 28, 36, 24, 25, 15, 8, 27, 45, 9, 21, 18

Offset: 1

Author

Zak Seidov, Nov 28 2012

Keywords

• nonn

Comments

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

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000 (First 1000 terms from Zak Seidov)
• Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications, Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.
• Robert Tijdeman, On integers with many small prime factors, Compos. Math. 26 (1973), 319-330.

Crossrefs

Cf. A051037, A085152.

Programs

Mathematica

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 *)

Formula

a(n) = A051037(n+1) - A051037(n).