A363788 -id:A363788 - cp's OEIS frontend

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

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));

A385482 a(n) is the least number k such that k*n is a binary Niven number (A049445).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 5, 1, 10, 3, 4, 1, 2, 1, 12, 1, 1, 3, 3, 1, 12, 5, 3, 3, 4, 2, 5, 1, 2, 1, 12, 1, 5, 6, 4, 1, 5, 1, 4, 3, 4, 2, 12, 1, 12, 6, 4, 3, 4, 2, 1, 3, 4, 2, 5, 1, 6, 5, 2, 1, 2, 1, 12, 1, 1, 6, 4, 1, 6, 3, 4, 3, 4, 2, 5, 1, 1, 3, 4, 1, 4

Offset: 1

Amiram Eldar, Jun 30 2025

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

Cf. A049445, A144261 (decimal analog), A363788, A385483 (indices of records), A385484 (record values), A385485.

a[n_] := Module[{m = n, k = 1}, While[!Divisible[m, DigitSum[m, 2]], m += n; k++]; k]; Array[a, 100]

a(n) = {my(m = n, k = 1); while(m % hammingweight(m), m += n; k++); k;}

from itertools import count
def a(n): return next(k for k in count(1) if (m:=k*n)%m.bit_count() == 0)
print([a(n) for n in range(1, 86)]) # Michael S. Branicky, Jun 30 2025

a(n) = 1 if and only if n is in A049445.

a(n) = 2 if and only if 2*n is in A363788.

cp's OEIS Frontend

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

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