A337078 -id:A337078 - cp's OEIS frontend

A386569 The number of binary self numbers not exceeding 2^n.

1, 1, 2, 3, 5, 10, 18, 34, 67, 131, 261, 520, 1037, 2073, 4143, 8283, 16562, 33121, 66237, 132471, 264938, 529870, 1059740, 2119473, 4238941, 8477878, 16955748, 33911492, 67822978, 135645949, 271291894, 542583782, 1085167557, 2170335106, 4340670206, 8681340402

Offset: 0

Amiram Eldar, Jul 26 2025

			There are 3 binary self numbers that do no exceed 2^3 = 8: 1, 4 and 6. Hence a(3) = 3.

Cf. A010061, A242403, A337078, A382452, A386568.

selfQ[n_] := AllTrue[Range[n, n - Floor@Log2[n], -1], # + DigitCount[#, 2, 1] != n &]; a[n_] := Count[Range[2^n], _?selfQ]; Array[a, 16, 0]

Limit_{n->oo} a(n)/2^n = A242403.

A337079 The number of twin binary Niven numbers (k, k+1) such that k <= 2^n.

Original entry on oeis.org

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

Amiram Eldar, Aug 14 2020

			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.

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.

Cf. A049445, A330931, A337078.

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

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.

cp's OEIS Frontend

A386569 The number of binary self numbers not exceeding 2^n.

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