A306263 -id:A306263 - cp's OEIS frontend

A306645 a(n) is the least positive multiple of n belonging to A306263 if any, or a(n) = -1 otherwise.

1, 2, 6, 4, 10, 6, 42, 8, 18, 10, 66, 12, 156, 42, 60, 16, 34, 18, 228, 20, 42, 66, 92, 24, 300, 156, 108, 84, 116, 60, 310, 32, 66, 34, 420, 36, 222, 228, 156, 40, 246, 42, 172, 132, 180, 92, 2820, 48, 588, 300, 204, 156, 212, 108, 660, 168, 228, 116, 590, 60

Offset: 1

Rémy Sigrist, Mar 03 2019

Is a(n) > 0 for every n > 0?

			For n = 7:
- the divisors of 7 are: 1, 7,
- the corresponding Hamming weights are: 1, 3,
- 3 does not divide 7,
- the divisors of 3*7 are: 1, 3, 7, 21,
- the corresponding Hamming weights are: 1, 2, 3, 3,
- 2 does not divide 3*7,
- the divisors of 2*3*7 are: 1, 2, 3, 6, 7, 14, 21, 42,
- the corresponding Hamming weights are: 1, 1, 2, 2, 3, 3, 3, 3,
- they all divide 2*3*7,
- hence a(7) = 2*3*7 = 42.

Cf. A000120, A306263.

With[{s = Select[Range[3000], With[{k = #}, AllTrue[Divisors@ k, Mod[k, DigitCount[#, 2, 1]] == 0 &]] &]}, Table[SelectFirst[s, Mod[#, n] == 0 &] /. k_ /; MissingQ@ k -> -1, {n, 60}]] (* Michael De Vlieger, Mar 05 2019 *)

a(n) = while (1, my (m=n); fordiv (m, d, m = lcm(m, hammingweight(d));); if (n==m, return (n), n = m))

a(A306263(n)) = A306263(n) for any n > 0.

A324393 a(n) is the number of such divisors d of n that A000120(d) does not divide n, where A000120(d) gives the binary weight of d.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 0, 1, 2, 1, 0, 2, 2, 3, 3, 1, 2, 1, 0, 2, 0, 3, 0, 1, 2, 2, 0, 1, 0, 1, 3, 5, 2, 1, 0, 2, 2, 3, 3, 1, 2, 2, 4, 2, 2, 1, 0, 1, 2, 3, 0, 3, 0, 1, 0, 2, 4, 1, 0, 1, 2, 4, 3, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 3, 4, 1, 4, 3, 0, 3, 2, 3, 0, 1, 4, 4, 3, 1, 2, 1, 4, 4

Offset: 1

Antti Karttunen, Mar 05 2019

Number of such positive integers k that divide n but A000120(k) [the Hamming weight of k] does not divide n.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A000005, A000120, A324392, A306263 (positions of zeros).

Cf. also A093653, A192895, A266342, A292257.

a[n_] := DivisorSum[n, 1 &, !Divisible[n, DigitCount[#, 2, 1]] &]; Array[a, 100] (* Amiram Eldar, Dec 04 2020 *)

A324393(n) = sumdiv(n, d, !!(n%hammingweight(d)));

a(n) = Sum_{d|n} [A000120(d) does not divide n], where [ ] is the Iverson bracket.
a(n) = A000005(n) - A324392(n).
a(p) = 1 for all odd primes p.

cp's OEIS Frontend

A306645 a(n) is the least positive multiple of n belonging to A306263 if any, or a(n) = -1 otherwise.

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