A182627 -id:A182627 - cp's OEIS frontend

A093653 Total number of 1's in binary expansion of all divisors of n.

1, 2, 3, 3, 3, 6, 4, 4, 5, 6, 4, 9, 4, 8, 9, 5, 3, 10, 4, 9, 9, 8, 5, 12, 6, 8, 9, 12, 5, 18, 6, 6, 8, 6, 9, 15, 4, 8, 10, 12, 4, 18, 5, 12, 15, 10, 6, 15, 7, 12, 9, 12, 5, 18, 11, 16, 10, 10, 6, 27, 6, 12, 17, 7, 8, 16, 4, 9, 10, 18, 5, 20, 4, 8, 16, 12, 11, 20, 6, 15, 12, 8, 5, 27, 9, 10, 12

Offset: 1

Jason Earls, May 16 2004

			a(8) = 4 because the divisors of 8 are [1, 2, 4, 8] and in binary: 1, 10, 100, 1000, so four 1's.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (first 500 terms from Jaroslav Krizek)
Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018. See (22) p. 6.
Index entries for sequences related to binary expansion of n

Cf. A000120, A093687, A192895, A292257.
Cf. A226590 (number of 0's in binary expansion of all divisors of n).
Cf. A182627 (number of digits in binary expansion of all divisors of n).
Cf. A034690 (a decimal equivalent).

a:= n-> add(add(i, i=Bits[Split](d)), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, May 17 2022

Table[Plus@@DigitCount[Divisors[n], 2, 1], {n, 75}] (* Alonso del Arte, Sep 01 2013 *)

A093653(n) = sumdiv(n,d,hammingweight(d)); \\ Antti Karttunen, Dec 14 2017

a(n) = {my(v = valuation(n, 2), n = (n>>v)); sumdiv(n, d, hammingweight(d)) * (v + 1)} \\ David A. Corneth, Feb 15 2023

from sympy import divisors
def a(n): return sum(bin(d).count("1") for d in divisors(n))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Apr 20 2022

from sympy import divisors
def A093653(n): return sum(d.bit_count() for d in divisors(n, generator=True))
print([A093653(n) for n in range(1, 88)]) # Michael S. Branicky, Feb 15 2023

a(n) = Sum_{k = 0..n} if(mod(n, k) = 0, A000120(k), 0). - Paul Barry, Jan 14 2005
a(n) = A182627(n) - A226590(n). - Jaroslav Krizek, Sep 01 2013
a(n) = A292257(n) + A000120(n). - Antti Karttunen, Dec 14 2017
From Bernard Schott, May 16 2022: (Start)
If prime p = A000043(n), then a(2^p-1) = a(A000668(n)) = p+1 = A050475(n).
a(2^n) = n+1 (End)

A182622 a(n) is the number whose binary representation is the concatenation of the divisors of n written in base 2.

Original entry on oeis.org

1, 6, 7, 52, 13, 222, 15, 840, 121, 858, 27, 28268, 29, 894, 991, 26896, 49, 113970, 51, 215892, 2037, 3446, 55, 14471576, 441, 3514, 3899, 217052, 61, 14538238, 63, 1721376, 7905, 13410, 7139, 926213284, 101, 13542, 8039, 221009192

Offset: 1

Omar E. Pol, Nov 22 2010

a(n) is A182621(n), interpreted as a binary number, written in base 10. The first repeated element is 991, from 15 and 479.

Except for 1, no power of 2 can occur in this sequence, an obvious consequence of the fact that a(n) has to be the sum of at least two distinct powers of 2 for all n > 1. - Alonso del Arte, Nov 13 2013

			The divisors of 10 are 1, 2, 5, 10. Then 1, 2, 5, 10 written in base 2 are 1, 10, 101, 1010. The concatenation of 1, 10, 101, 1010 is 1101011010. Then a(10) = 858 because the binary number 1101011010 written in base 10 is 858.

Indranil Ghosh, Table of n, a(n) for n = 1..50000

Cf. A027750, A007088, A182620, A182621, A182623, A182624, A182627, A182632.

concatBits[n_] := FromDigits[Join @@ (IntegerDigits[#, 2]& /@ Divisors[n]), 2]; concatBits /@ Range[40](* Giovanni Resta, Nov 23 2010 *)

a(n) = {my(cbd = []); fordiv(n, d, cbd = concat(cbd, binary(d));); fromdigits(cbd, 2);} \\ Michel Marcus, Jan 28 2017

def A182622(n):
    s=""
    for i in range(1,n+1):
        if n%i==0:
            s+=bin(i)[2:]
    return int(s,2) # Indranil Ghosh, Jan 28 2017

a(p) = 2^(floor(log_2(p)) + 1) + p for p prime. Also, a(p + k) > a(p) for all k > 0. Furthermore, for all primes p > 3, a(p) < a(p - 1).

a(2^(m - 1)) = sum(k = 0 .. m - 1, 2^((m^2 + m)/2 - (k^2 + k)/2 - 1)) = A164894(m). - Alonso del Arte, Nov 13 2013

More terms from Giovanni Resta, Nov 23 2010

A226590 Total number of 0's in binary expansion of all divisors of n.

Original entry on oeis.org

0, 1, 0, 3, 1, 2, 0, 6, 2, 4, 1, 6, 1, 2, 1, 10, 3, 7, 2, 9, 2, 4, 1, 12, 3, 4, 3, 6, 1, 6, 0, 15, 5, 8, 4, 15, 3, 6, 3, 16, 3, 8, 2, 9, 5, 4, 1, 20, 3, 9, 5, 9, 2, 10, 3, 12, 4, 4, 1, 15, 1, 2, 4, 21, 7, 14, 4, 15, 5, 12, 3, 26, 4, 8, 6, 12, 4, 10, 2, 25, 7

Offset: 1

Jaroslav Krizek, Aug 31 2013

Also total number of 0's in binary expansion of concatenation of the binary numbers that are the divisors of n written in base 2 (A182621).

a(n) = 0 iff n = 1 or n is a Mersenne prime (A000668). - Bernard Schott, Apr 22 2022

			a(8) = 6 because the divisors of 8 are [1, 2, 4, 8] and in binary: 1, 10, 100, 1000, so six 0's.

Jaroslav Krizek, Table of n, a(n) for n = 1..500

Cf. A093653 (number of 1's in binary expansion of all divisors of n).
Cf. A182627 (number of digits in binary expansion of all divisors of n).
Cf. A182621 (concatenation of the divisors of n written in base 2).
Cf. A000217, A000668.

Table[Count[Flatten[IntegerDigits[Divisors[n], 2]], 0], {n, 81}] (* T. D. Noe, Sep 04 2013 *)

a(n) = sumdiv(n, d, 1+logint(d, 2) - hammingweight(d)); \\ Michel Marcus, Apr 24 2022

from sympy import divisors
def a(n): return sum(bin(d)[2:].count("0") for d in divisors(n))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Apr 20 2022

a(n) = A182627(n) - A093653(n).

a(2^n) = n*(n+1)/2 = A000217(n). - Bernard Schott, Apr 22 2022

A182628 Triangle T(n,k) read by rows in which row n lists the number of digits of the binary expansion of the divisors of n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 23 2010

Row n lists the number of digits of the numbers in the n-th row of triangle A182620.

			Triangle begins:
1,
1, 2,
1, 2,
1, 2, 3,
1, 3,
1, 2, 2, 3,
1, 3,
1, 2, 3, 4,
1, 2, 4,
1, 2, 3, 4,
1, 4,
1, 2, 2, 3, 3, 4,

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A182620, A182621, A182622.

Row sums give A182627.

with(numtheory):for n from 1 to 12 do for d in divisors(n) do printf("%d, ",length(convert(convert(d, binary),string))); od:printf("\n"); od: # Nathaniel Johnston, Apr 19 2011

a(38)-a(79) from Nathaniel Johnston, Apr 19 2011

A231116 Numbers k such that the total number of digits of all the divisors of k written in base 2 is equal to k.

Original entry on oeis.org

1, 3, 10, 24

Offset: 1

Jaroslav Krizek, Nov 03 2013

Sequence is finite with 4 terms.
Numbers k such that the concatenation of their divisors written in base 2 contains k digits.
Subsequence of finite sequence:  1, 2, 3, 4, 6, 8, 10, 12, 24 (numbers k such that the total number of digits of all the divisors of k written in base 2 is greater than or equal to k).
Numbers k such that A182627(k) = k.

			24 is in the sequence because the concatenation of the divisors of 24 in base 2 [(1, 10, 11, 100, 110, 1000, 1100, 11000) -> 110111001101000110011000] contains 24 digits.

Cf. A182622, A182627.

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A093653 Total number of 1's in binary expansion of all divisors of n.

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A182622 a(n) is the number whose binary representation is the concatenation of the divisors of n written in base 2.

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A226590 Total number of 0's in binary expansion of all divisors of n.

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Mathematica

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A182628 Triangle T(n,k) read by rows in which row n lists the number of digits of the binary expansion of the divisors of n.

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Maple

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A231116 Numbers k such that the total number of digits of all the divisors of k written in base 2 is equal to k.

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