A352086 -id:A352086 - cp's OEIS frontend

A352084 Integers m such that wt(m) divides wt(m^2) where wt(m) = A000120(m) is the binary weight of m.

1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 21, 24, 28, 30, 31, 32, 37, 42, 45, 48, 53, 56, 60, 62, 63, 64, 69, 73, 74, 79, 81, 83, 84, 90, 91, 96, 106, 112, 120, 124, 126, 127, 128, 133, 137, 138, 141, 146, 148, 155, 157, 158, 159, 161, 162, 165, 166, 168, 177, 180

Offset: 1

Bernard Schott, Mar 03 2022

Integers m such that A000120(m) divides A159918(m).
This is a problem proposed by the French site Diophante in the links section.
The first 18 terms are the same as A268415, then A268415(19) = 41 while a(19) = 42.
The corresponding quotients are in A352085.
The smallest term k such that the corresponding quotient = n is A352086(n).
Some subsequences:
-> wt(m^2) = wt(m) iff m is in A077436.
-> wt(m^2) / wt(m) = 2 iff m is in A083567.
-> When m is a power of 2 (A000079): wt(2^k) = wt((2^k)^2) = wt(2^(2k)) = 1.

			37_10 = 100101_2, digsum_2(37) = 1+1+1 = 3; then 37^2 = 1369_10 = 10101011001_2, digsum_2(1369) = 1+1+1+1+1+1 = 6; as 3 divides 6, 37 is a term.

Martin Ehrenstein, Table of n, a(n) for n = 1..10000
Diophante, A1730 - Des chiffres à sommer pour un entier (in French).

Cf. A000120, A159918, A268415, A351650, A352085, A352086.
Cf. A351650 (similar for base 10).
Subsequences: A000079, A023758, A077436, A083567.

Select[Range[180], Divisible[Total[IntegerDigits[#^2, 2]], Total[IntegerDigits[#, 2]]] &] (* Amiram Eldar, Mar 03 2022 *)

isok(m) = !(hammingweight(m^2) % hammingweight(m)); \\ Michel Marcus, Mar 03 2022

def ok(n): return n > 0 and bin(n**2).count('1')%bin(n).count('1') == 0
print([m for m in range(1, 200) if ok(m)]) # Michael S. Branicky, Mar 03 2022

More terms from Amiram Eldar, Mar 03 2022

A352085 a(n) is the quotient wt(m^2) / wt(m), where wt = binary weight = A000120 and m = A352084(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2

Offset: 1

Bernard Schott, Mar 05 2022

All positive integers are terms of this sequence and the smallest integer k such that a(k) = n is A352086(n).

a(n) = 1 iff m = A352084(n) is a term of A077436, and a(n) = 2 iff m = A352084(n) is a term of A083567.

			For n=19, A352084(19) = 42_10 = 101010_2 => wt(42) = 3 ones; then 42^2 = 1764_10 = 11011100100_2 => wt(1764) = 6 ones, so that a(19) = wt(42^2) / wt(42) = 6/3 = 2.

Martin Ehrenstein, Table of n, a(n) for n = 1..10000
Diophante, A1730 - Des chiffres à sommer pour un entier (in French).

Cf. A000120, A077436, A083567, A159918, A352084, A352086.

Select[Total[IntegerDigits[#^2, 2]]/Total[IntegerDigits[#, 2]] & /@ Range[300], IntegerQ] (* Amiram Eldar, Mar 05 2022 *)

lista(nn) = my(list = List(), q); for (n=1, nn, if (denominator(q=hammingweight(n^2)/hammingweight(n)) == 1, listput(list, q));); Vec(list); \\ Michel Marcus, Mar 05 2022

from itertools import count, islice
def agen(): # generator of terms
    for m in count(1):
        q, r = divmod(bin(m**2).count('1'), bin(m).count('1'))
        if r == 0:
            yield q
print(list(islice(agen(), 100))) # Michael S. Branicky, Mar 05 2022

a(n) = A159918(A352084(n))/A000120(A352084(n)).

More terms from Michel Marcus, Mar 05 2022

cp's OEIS Frontend

A352084 Integers m such that wt(m) divides wt(m^2) where wt(m) = A000120(m) is the binary weight of m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions