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

1,6

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.

Formula

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

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 *)

Crossrefs

Cf. A051037, A085152.

Sequence in context: A181118 A179009 A112757 * A351469 A286334 A118492

Adjacent sequences: A219791 A219792 A219793 * A219795 A219796 A219797

Keyword

nonn

Author

Zak Seidov, Nov 28 2012

Status

approved