A337079 The number of twin binary Niven numbers (k, k+1) such that k <= 2^n.
1, 1, 1, 1, 2, 2, 5, 8, 18, 35, 61, 98, 187, 304, 492, 880, 1583, 2779, 5196, 9407, 17387, 31772, 58450, 106360, 193875, 351836, 642844, 1173333, 2155913, 3993379, 7466547, 14048253, 26680668, 50751057, 97052665, 185557893, 354235368, 674995568, 1284856970
Offset: 1
Examples
a(5) = 2 since there are two binary Niven numbers k below 2^5 = 32 such that k+1 is also a binary Niven number: 1 and 20.
Links
- Jean-Marie De Koninck, Nicolas Doyon and Imre Kátai, Counting the number of twin Niven numbers, The Ramanujan Journal, Vol. 17, No .1 (2008), pp. 89-105, alternative link.
Programs
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Mathematica
binNivenQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; s = {}; c = 0; p = 2; q1 = True; Do[q2 = binNivenQ[n]; If[q1 && q2, c++]; If[n - 1 == p, AppendTo[s, c]; p *= 2]; q1 = q2, {n, 2, 2^20}]; s
Formula
a(n) ~ c * 2^n/n^2, where c is a constant (consequence of the theorem of De Koninck et al., 2008). Apparently c ~ 0.28.