A144302 -id:A144302 - cp's OEIS frontend

A376618 Odd binary Niven numbers (A144302) k such that k/wt(k) is also an odd binary Niven number, where wt(k) = A000120(k) is the binary weight of k.

1, 345, 405, 775, 1305, 1425, 1435, 1605, 2125, 2325, 2485, 2765, 2825, 4235, 4305, 4459, 4655, 4725, 5085, 5145, 5607, 5625, 5929, 6223, 6405, 7515, 7623, 8145, 10625, 11151, 11835, 12325, 12355, 12425, 13527, 13825, 13995, 14805, 16695, 18445, 20505, 20625, 20925

Offset: 1

Amiram Eldar, Sep 30 2024

If m is a term then 2^k * m is a term of A376616 for all k >= 0.

			345 is a term since it is odd, 345/wt(345) = 69 is an integer, and 69/wt(69) = 23 is an integer.

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

Intersection of A005408 and A376616.
Subsequence of A049445 and A144302.
Cf. A000120.

q[k_] := Module[{w = DigitCount[k, 2, 1]}, Divisible[k, w] && Divisible[k/w, DigitCount[k/w, 2, 1]]]; Select[Range[1, 21000, 2], q]

is(k) = if(!(k % 2), 0, my(w = hammingweight(k)); !(k % w) && !((k/w) % hammingweight(k/w)));

A363787 Primitive binary Niven numbers: binary Niven numbers (A049445) that are not twice another binary Niven number.

Original entry on oeis.org

1, 6, 10, 18, 21, 34, 55, 60, 66, 69, 81, 92, 108, 115, 116, 126, 130, 155, 156, 172, 180, 185, 204, 205, 212, 222, 228, 246, 258, 261, 273, 284, 285, 295, 300, 308, 318, 321, 332, 340, 345, 355, 356, 366, 378, 395, 396, 404, 405, 414, 420, 425, 438, 452, 462

Offset: 1

Amiram Eldar, Jun 22 2023

Every binary Niven number is of the form m*2^k, where m is a term of this sequence and k >= 0.
Includes all the odd binary Niven numbers (A144302).
This sequence is infinite. E.g., 16^k + 4^k + 1 is a term for all k >= 1.

			6 is a term as 6 is a binary Niven number and 6/2 = 3 is not a binary Niven number.

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

Subsequence of A049445.
Disjoint union of A144302 and A363788.
A363789 is a subsequence.
Cf. A356349 (decimal analog).

binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; q[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]); Select[Range[500], q]

isbinniv(n) = !(n % hammingweight(n));
is(n) = isbinniv(n) && !(!(n%2) && isbinniv(n/2));

A363788 Even primitive binary Niven numbers: even terms of A363787.

Original entry on oeis.org

6, 10, 18, 34, 60, 66, 92, 108, 116, 126, 130, 156, 172, 180, 204, 212, 222, 228, 246, 258, 284, 300, 308, 318, 332, 340, 356, 366, 378, 396, 404, 414, 420, 438, 452, 462, 474, 486, 498, 514, 540, 556, 564, 588, 596, 606, 612, 630, 652, 660, 676, 708, 726, 780

Offset: 1

Amiram Eldar, Jun 22 2023

The odd terms of A363787 are all the odd binary Niven numbers (A144302).

This sequence is infinite. E.g., A052548(k) = 2^k + 2 is a term for all k >= 2.

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

Subsequence of A049445 and A363787.
Equals A363787 \ A144302.
Cf. A052548, A358255 (decimal analog).

binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; q[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]); Select[Range[2, 1000, 2], q]

isbinniv(n) = !(n % hammingweight(n));
is(n) = !(n%2) && isbinniv(n) && !isbinniv(n/2);

A376619 a(n) is the least odd number k such that A376615(k) = n, or -1 if no such number exists.

Original entry on oeis.org

3, 21, 345, 10625, 74375, 860625, 84189105, 1599592995, 23993894925

Offset: 1

Amiram Eldar, Sep 30 2024

Without the restriction to odd numbers the corresponding sequence is 3*2^(n-1) = A007283(n-1).
All the terms above 3 are odd binary Niven numbers (A144302).
a(10) > 10^13, if it exists.

			  n | The n iterations
  --+------------------------------------------------------
  1 | 3 -> 3/2
  2 | 21 -> 7 -> 7/3
  3 | 345 -> 69 -> 23 -> 23/4
  4 | 10625 -> 2125 -> 425 -> 85 -> 85/4
  5 | 74375 -> 10625 -> 2125 -> 425 -> 85 -> 85/4
  6 | 860625 -> 95625 -> 10625 -> 2125 -> 425 -> 85 -> 85/4

Cf. A007283, A049445, A144302, A376615, A376616, A376617.

s[n_] := s[n] = Module[{bw = DigitCount[n, 2, 1]}, If[bw == 1, 0, If[!Divisible[n, bw], 1, 1 + s[n/bw]]]]; seq[len_] := Module[{v = Table[0, {len}], c = 0, k = 3, i}, While[c < len, i = s[k]; If[v[[i]] == 0, c++; v[[i]] = k]; k += 2]; v]; seq[5]

s(n) = {my(w = hammingweight(n)); if(w == 1, 0, if(n % w, 1, 1 + s(n/w)));}
lista(len) = {my(v = vector(len), c = 0, k = 3, i); while(c < len, i = s(k); if(v[i] == 0, c++; v[i] = k); k += 2); v;}

A152567 Numbers k such that A049445(k) is odd.

Original entry on oeis.org

1, 11, 19, 24, 27, 33, 43, 51, 54, 68, 71, 74, 76, 83, 89, 90, 98, 101, 107, 117, 130, 135, 138, 144, 151, 153, 156, 163, 165, 178, 181, 188, 195, 199, 203, 205, 207, 212, 215, 226, 230, 235, 238, 244, 249, 251, 258, 267, 272, 278, 282, 285, 294, 298, 304, 305, 325, 327

Offset: 1

Roger L. Bagula, Dec 07 2008

nonn

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

Cf. A005349, A049445, A144302.

Position[Select[Range[1, 1500], Divisible[#, DigitCount[#, 2, 1]] &], ?OddQ] // Flatten (* _Amiram Eldar, Aug 16 2020 *)

Edited by N. J. A. Sloane, Dec 07 2008

cp's OEIS Frontend

A376618 Odd binary Niven numbers (A144302) k such that k/wt(k) is also an odd binary Niven number, where wt(k) = A000120(k) is the binary weight of k.

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A363787 Primitive binary Niven numbers: binary Niven numbers (A049445) that are not twice another binary Niven number.

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A363788 Even primitive binary Niven numbers: even terms of A363787.

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A376619 a(n) is the least odd number k such that A376615(k) = n, or -1 if no such number exists.

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A152567 Numbers k such that A049445(k) is odd.

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