A337077 Binary Niven numbers (A049445) with a record gap to the next binary Niven number.

1, 2, 12, 24, 96, 690, 1386, 3024, 3738, 3794, 5544, 22834, 57278, 68908, 89060, 196240, 360000, 388421, 524160, 1556360, 1572480, 2359140, 3929940, 8057711, 11484900, 15201585, 16115505, 19910436, 32444160, 7348411575, 16097143458, 33273395232, 51333952011

Offset: 1

Author

Amiram Eldar, Aug 14 2020

Keywords

• nonn
• base

Comments

The corresponding record gaps are 1, 2, 4, 8, 12, 18, 26, 27, 33, 38, 42, 44, 46, 50, 58, 68, 74, 77, 103, 109, 122, 137, 156, 157, 165, 189, 191, 204, 240, 265, 267, 312, 333, ...

De Koninck, Doyon and Kátai (2003) proved that the asymptotic density of the Niven numbers in any base >= 2 is 0. Therefore, the asymptotic density of the binary Niven numbers is 0 and this sequence is infinite.

Examples

			The first 8 binary Niven numbers are 1, 2, 4, 6, 8, 10, 12 and 16. The differences between them are 1, 2, 2, 2, 2, 2 and 4. The record gaps, 1, 2 and 4, occur at 1, 2 and 12.
		

Links

• Jean-Marie De Koninck and Nicolas Doyon, Large and Small Gaps Between Consecutive Niven Numbers, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.
• Jean-Marie De Koninck, Nicolas Doyon and Imre Kátai, On the counting function for the Niven numbers, Acta Arithmetica, Vol. 106, No. 3 (2003), 265-275.

Crossrefs

Cf. A049445, A330931, A337076.

Programs

Mathematica

binNivenQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; gapmax = 0; n1 = 1; s = {}; Do[If[binNivenQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, 10^6}]; s