A093687 -id:A093687 - 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)

A195307 Where records occur in A129308 and also in A195155.

Original entry on oeis.org

1, 2, 6, 12, 60, 180, 360, 420, 840, 1260, 2520, 5040, 13860, 27720, 55440, 83160, 166320, 277200, 360360, 720720, 1081080, 2162160, 2827440, 4324320, 6126120, 12252240, 24504480, 36756720, 73513440, 147026880, 183783600, 232792560, 367567200, 465585120, 698377680

Offset: 1

Omar E. Pol, Oct 16 2011

nonn

Observation: a(n) ending at 0, if 5 <= n <= 24 and possibly more.
From David A. Corneth, Apr 14 2021: (Start)
Conjecture: for each term k > 1 in the sequence there exists prime p such that k/p is in the sequence.
From the first 35 terms only a(23) = 2827440 is not in A025487.
In the list of conjectured terms, if actual terms <= 10^16 are 97-smooth and have the following property: a(n+1) = a(n) + k*gcd(a(n), a(n-1), ..., a(n-20)) setting a(n) = 1 for n < 1 then those terms are actual terms.
The conjectured terms are 41-smooth and satisfy a(n+1) = a(n) + k*gcd(a(n), a(n-1), ..., a(n-13)). (End)
From Bernard Schott, Jul 30 2022: (Start)
Equivalently, integers whose number of oblong divisors (A129308) sets a new record.
Corresponding records of number of oblong divisors are 0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ... (End)

			a(4) = 12 is in the sequence because A129308(12) = 3 is larger than any earlier value in A129308. - _Bernard Schott_, Jul 30 2022

David A. Corneth, Conjectured terms <= 10^16 (a(1)..a(35) are certain)

Cf. A002378, A007862, A088726, A093687, A097212, A129308, A181826, A195155, A287142.

More terms a(6)-a(24) from Alois P. Heinz, Oct 16 2011

a(25)-a(35) from David A. Corneth, Apr 14 2021

A360641 Numbers k where A093653(k)/A000120(k) sets a new record.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 36, 66, 72, 132, 144, 264, 420, 528, 840, 1026, 1056, 1680, 2052, 4104, 8208, 16416, 32832, 65664, 73920, 84000, 110880, 118800, 131328, 133380, 237600, 263340, 266760, 526680, 533520, 1053360, 1067040, 2106720, 2134080, 3160080, 4213440

Offset: 1

Amiram Eldar, Feb 15 2023

Analogous to superabundant numbers (A004394) as A175526 is analogous to abundant numbers (A005101).
The corresponding record values are 1, 2, 3, 4, 9/2, 5, 6, 15/2, 8, 10, ... .
This sequence is infinite since A093653(k)/A000120(k) is unbounded: A093653(2^m)/A000120(2^m) = m+1 for all m >= 0.

			The values of A093653(k)/A000120(k) for k=1..10 are 1, 2, 3/2, 3, 3/2, 3, 4/3, 4, 5/2 and 3. The record values, 1, 2, 3 and 4, occur at 1, 2, 4 and 8, the first 4 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..58 (terms below 10^10)

Cf. A000120, A004394, A005101, A093653, A093687, A175522, A175526, A360642.

seq[nmax_] := Module[{s = {}, rm = 0, r}, Do[If[(r = DivisorSum[n, DigitCount[#, 2, 1] &]/DigitCount[n, 2, 1]) > rm, rm = r; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[10^4]

lista(kmax) = {my(rm = 0, r); for(k = 1, kmax, r = sumdiv(k, d, hammingweight(d))/hammingweight(k); if(r > rm, rm = r; print1(k, ", "))); }

# uses imports and definitions in A093653, A000120
from itertools import count, islice
def f(n): return A093653(n)/A000120(n)
def agen(r=0): yield from ((m, r:=fm)[0] for m in count(1) if (fm:=f(m))>r)
print(list(islice(agen(), 34))) # Michael S. Branicky, Feb 15 2023

A360642 a(n) is the least number k such that A093653(k)/A000120(k) = n.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 64, 66, 84, 72, 210, 132, 450, 792, 288, 264, 1044, 672, 5328, 528, 1344, 840, 1026, 1056, 4116, 1800, 4128, 2112, 5124, 3780, 6480, 2184, 3360, 8352, 11088, 8448, 4680, 50700, 4200, 4368, 20880, 8280, 13320, 13440, 12420, 4104, 46200, 8736

Offset: 1

Amiram Eldar, Feb 15 2023

a(n) exists for all n >= 1 since A093653(2^(k-1))/A000120(2^(k-1)) = k for all k >= 1.

Analogous to A007539 as A175522 is analogous to perfect numbers (A000396).

			a(1) = 1 since A093653(1)/A000120(1) = 1/1 = 1.
a(2) = 2 since A093653(2)/A000120(2) = 2/1 = 2, and 2 is the least number with this property.
a(3) = 4 since A093653(4)/A000120(4) = 3/1 = 3, and 4 is the least number with this property.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000120, A000396, A007539, A093653, A093687, A175522, A175526, A360641.

seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n <= nmax, i = DivisorSum[n, DigitCount[#, 2, 1] &]/DigitCount[n, 2, 1]; If[IntegerQ[i] && i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[50, 10^5]

lista(len, nmax) = {my(s = vector(len), c = 0, n = 1, i); while(c < len && n <= nmax, i = sumdiv(n, d, hammingweight(d))/hammingweight(n); if(denominator(i) == 1 && i <= len && s[i] == 0, c++; s[i] = n); n++); s }

# uses imports and definitions in A093653, A000120
from itertools import count, islice
def f(n): q, r = divmod(A093653(n), A000120(n)); return q if r == 0 else 0
def agen():
    n, adict = 1, dict()
    for k in count(1):
        v = f(k)
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 48))) # Michael S. Branicky, Feb 15 2023

a(n) <= 2^(n-1).

A339552 Numbers k such that the product of the binary weights of the divisors of k (A339549) sets a new record.

Original entry on oeis.org

1, 3, 6, 12, 14, 15, 21, 28, 30, 42, 60, 84, 90, 120, 168, 180, 210, 252, 360, 420, 540, 630, 840, 1080, 1260, 2520, 3780, 5040, 6300, 7560, 10080, 12600, 13860, 15120, 21420, 22680, 25200, 27720, 32760, 37800, 41580, 42840, 49140, 55440, 65520, 75600, 83160

Offset: 1

Amiram Eldar, Dec 08 2020

Analogous to A093687 as A339549 is analogous to A093653.

The corresponding record values of A339549 are 1, 2, 4, 8, 9, 16, 18, 27, 256, 324, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..150
Amiram Eldar, Table of n, a(n), A339549(a(n)) for n = 1..150

Cf. A000120, A093653, A093687, A339549.

f[n_] := Times @@ (DigitCount[#, 2, 1] & /@ Divisors[n]); c=0; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10000}]; s

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

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A195307 Where records occur in A129308 and also in A195155.

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A360641 Numbers k where A093653(k)/A000120(k) sets a new record.

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A360642 a(n) is the least number k such that A093653(k)/A000120(k) = n.

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A339552 Numbers k such that the product of the binary weights of the divisors of k (A339549) sets a new record.

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