A330931 -id:A330931 - cp's OEIS frontend

A331821 Positive numbers k such that -k and -(k + 1) are both negabinary-Niven numbers (A331728).

2, 3, 8, 9, 15, 24, 27, 32, 33, 39, 54, 55, 63, 77, 111, 114, 115, 123, 128, 129, 135, 144, 159, 174, 175, 203, 234, 235, 245, 255, 264, 294, 295, 329, 370, 371, 384, 413, 414, 415, 444, 447, 474, 475, 495, 504, 507, 512, 513, 519, 534, 535, 543, 580, 581, 624

Offset: 1

Amiram Eldar, Jan 27 2020

			8 is a term since both -8 and -(8 + 1) = -9 are negabinary-Niven numbers: A039724(-8) = 1000 and 1 + 0 + 0 + 0 = 1 is a divisor of 8, and A039724(-9) = 1011 and 1 + 0 + 1 + 1 = 3 is a divisor of 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005352, A027615, A039724, A328209, A328213, A330927, A330931, A331086, A331089, A331728, A331819, A331820.

negaBinWt[n_] := negaBinWt[n] = If[n == 0, 0, negaBinWt[Quotient[n - 1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[-n]]; c = 0; k = 1; s = {}; v = Table[-1, {2}]; While[c < 60, If[negaBinNivenQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 1]]]; k++]; s

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

A331821 Positive numbers k such that -k and -(k + 1) are both negabinary-Niven numbers (A331728).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica