A093653 -id:A093653 - cp's OEIS frontend

A385494 Total number of 1's in the decimal digits of the divisors of n.

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

Offset: 1

Robert Israel, Aug 27 2025

			a(11) = 3 because of the divisors of 11, there is one 1 in 1 and two in 11.
a(60) = 4 because of the divisors of 60, there is one 1 in 1, one in 10, one in 12, one in 15 and none in the other divisors.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A093653, A387357.

f:= proc(n) local d; add(numboccur(1, convert(d,base,10)),d=numtheory:-divisors(n)) end proc:
map(f, [$1..100]);

a[n_]:=Count[IntegerDigits[Divisors[n]]//Flatten,1]; Array[a,100] (* Stefano Spezia, Aug 28 2025 *)

a(n) = sumdiv(n, d, #select(x->(x==1), digits(d))); \\ Michel Marcus, Aug 28 2025

from sympy import divisors
def a(n): return sum(str(d).count("1") for d in divisors(n, generator=True))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Aug 27 2025

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

A318448 a(n) = Sum_{d|n} A294898(d), where A294898(d) = A005187(d) - sigma(d).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 2, 7, -8, 9, 4, 4, 0, 14, -4, 15, -2, 10, 12, 18, -22, 18, 16, 13, 1, 24, -14, 25, 0, 23, 26, 24, -31, 33, 28, 27, -14, 37, -6, 38, 13, 15, 34, 41, -52, 41, 22, 40, 19, 48, -10, 42, -10, 45, 46, 53, -76, 55, 48, 29, 0, 55, 12, 63, 34, 57, 18, 66, -98, 69, 64, 42, 37, 64, 16, 73, -42, 51, 72, 78, -74, 74, 74, 73, 6

Offset: 1

Antti Karttunen, Aug 27 2018

sign

Inverse Möbius transform of A294898.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A005187, A007429, A093653, A294898, A296075, A318446, A318447.

Cf. also A297114, A317844.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n) - sigma(n));
A318448(n) = sumdiv(n,d,A294898(d));

a(n) = Sum_{d|n} A294898(d).
a(n) = A318447(n) + A294898(n).
a(n) = A318446(n) - A007429(n).
a(n) = A296075(n) - A093653(n).

A324392 a(n) is the number of divisors d of n such that A000120(d) divides n, where A000120(d) gives the binary weight of d.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 4, 1, 4, 1, 6, 1, 2, 1, 5, 1, 6, 1, 6, 3, 2, 1, 8, 1, 2, 1, 3, 1, 6, 1, 6, 2, 4, 1, 9, 1, 2, 2, 8, 1, 8, 1, 3, 1, 2, 1, 10, 1, 4, 1, 3, 1, 6, 2, 4, 2, 2, 1, 12, 1, 2, 3, 7, 1, 8, 1, 6, 2, 4, 1, 12, 1, 2, 2, 3, 1, 6, 1, 10, 2, 2, 1, 12, 1, 2, 1, 4, 1, 8, 1, 6, 1, 2, 1, 12, 1, 2, 2, 6, 1, 6, 1, 4, 4

Offset: 1

Antti Karttunen, Mar 05 2019

Number of such positive integers k that both k and A000120(k) [the Hamming weight of k] 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, A324393.

Cf. also A093653, A192895, A266342, A292257.

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

A324392(n) = sumdiv(n, d, !(n%hammingweight(d)));

a(n) = Sum_{d|n} [A000120(d) does divide n], where [ ] is the Iverson bracket.
a(n) = A000005(n) - A324393(n).
a(p) = 1 for all odd primes p.

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.

A333618 a(n) is the total number of terms (1-digits) in the dual Zeckendorf representation of all divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 3, 7, 3, 7, 6, 7, 5, 12, 4, 8, 8, 11, 5, 14, 6, 12, 9, 10, 5, 20, 7, 9, 11, 14, 6, 20, 6, 17, 11, 10, 10, 23, 6, 12, 11, 21, 5, 22, 6, 17, 17, 11, 6, 30, 8, 17, 13, 17, 8, 23, 12, 22, 13, 13, 6, 33, 7, 12, 18, 23, 12, 26, 6, 17, 13, 23, 7, 37, 7, 14

Offset: 1

Amiram Eldar, Mar 29 2020

			For n = 6, its divisors are 1, 2, 3 and 6. The dual Zeckendorf representations (A104326) of the divisors are 1, 10, 11 and 111. Their total number of 1's is 1 + 1 + 2 + 3 = 7, thus a(6) = 7.

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

Cf. A034690, A093653, A104326, A112310, A300837.

fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr];
dualZeckSum[n_] := Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]];
a[n_] := DivisorSum[n, dualZeckSum[#] &]; Array[a, 100]

a(n) = Sum_{d|n} A112310(d).

A339550 Numbers k such that A339549(k) = A339549(k+1).

Original entry on oeis.org

1, 9, 85, 697, 1285, 2605, 4573, 5845, 6001, 6241, 6613, 7141, 7453, 8005, 10897, 12453, 13141, 15445, 19789, 20345, 21445, 21913, 22873, 25957, 36565, 36601, 39597, 44761, 46405, 53677, 56137, 56593, 61013, 63445, 70094, 72913, 76977, 80913, 82405, 87085, 87601

Offset: 1

Amiram Eldar, Dec 08 2020

Analogous to A338452 as A339549 is analogous to A093653.

			9 is a term since A339549(9) = A339549(10) = 4.

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

Cf. A000120, A093653, A338452, A339549.

f[n_] := Times @@ (DigitCount[#, 2, 1] & /@ Divisors[n]); Select[Range[10000], f[#] == f[# + 1] &]

A360639 Numbers k such that k and k+2 are both A000120-perfect numbers (A175522).

Original entry on oeis.org

123, 219, 695, 1261, 1851, 1943, 3543, 5963, 7031, 7613, 7769, 7861, 10081, 11357, 11629, 12083, 13211, 13791, 14185, 15699, 15835, 15929, 16241, 18649, 20197, 20989, 22521, 23449, 23521, 23963, 24461, 27215, 27829, 28263, 28367, 29485, 29651, 30359, 30901, 31803

Offset: 1

Amiram Eldar, Feb 15 2023

The smallest gap between two consecutive A000120-perfect numbers is 2.

All terms of this sequence are odd.

			123 is a term since 123 and 125 are both in A175522: A093653(123)/A000120(123) = A093653(125)/A000120(125) = 12/6 = 2.

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

Subsequence of A175522.

Cf. A000120, A093653, A338452, A360640.

q[n_] := DivisorSum[n, DigitCount[#, 2, 1] &] == 2 * DigitCount[n, 2, 1]; seq[kmax_] := Module[{s = {}, k = 1, q1 = False, q2}, Do[q2 = q[k]; If[q1 && q2, AppendTo[s, k-2]]; q1 = q2, {k, 3, kmax, 2}]; s]; seq[32000]

lista(kmax) = {my(is1 = 0, is2); forstep(k=1, kmax, 2, is2 = (sumdiv(k, d, hammingweight(d)) == 2*hammingweight(k)); if(is1 && is2, print1(k-2, ", ")); is1 = is2); }

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

A354112 Total number of 1's in binary expansion of all divisors of 2^n-1.

Original entry on oeis.org

1, 3, 4, 9, 6, 17, 8, 27, 16, 33, 20, 100, 14, 44, 42, 81, 18, 186, 20, 293, 80, 118, 38, 634, 62, 77, 64, 523, 80, 813, 32, 243, 153, 99, 154, 5031, 58, 110, 189, 1918, 67, 1624, 115, 1545, 761, 226, 120, 9366, 64, 1728, 472, 1861, 135, 2162, 945, 3471, 261, 1056, 101, 73418

Offset: 1

Michel Marcus, May 17 2022

Michel Marcus, Table of n, a(n) for n = 1..300

Cf. A000225, A093653.

a:= n-> add(add(i, i=Bits[Split](d)), d=numtheory[divisors](2^n-1)):
seq(a(n), n=1..60);  # Alois P. Heinz, May 17 2022

a[n_] := Total[DigitCount[Divisors[2^n - 1], 2, 1]]; Array[a, 60] (* Amiram Eldar, May 17 2022 *)

a(n) = sumdiv(2^n-1, d, hammingweight(d));

# if python version < 3.10, replace d.bitcount() with bin(d).count('1')
from sympy import divisors
def A354112(n): return sum(d.bit_count() for d in divisors(2**n-1,generator=True)) # Chai Wah Wu, May 17 2022

a(n) = A093653(A000225(n)).

cp's OEIS Frontend

A385494 Total number of 1's in the decimal digits of the divisors of n.

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

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A318448 a(n) = Sum_{d|n} A294898(d), where A294898(d) = A005187(d) - sigma(d).

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A324392 a(n) is the number of divisors d of n such that A000120(d) divides n, where A000120(d) gives the binary weight of d.

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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.

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A333618 a(n) is the total number of terms (1-digits) in the dual Zeckendorf representation of all divisors of n.

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A339550 Numbers k such that A339549(k) = A339549(k+1).

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A360639 Numbers k such that k and k+2 are both A000120-perfect numbers (A175522).

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