A046801 -id:A046801 - cp's OEIS frontend

A361438 Triangle T(n,k), n >= 1, 1 <= k <= A046801(n), read by rows, where T(n,k) is k-th smallest divisor of 2^n-1.

1, 1, 3, 1, 7, 1, 3, 5, 15, 1, 31, 1, 3, 7, 9, 21, 63, 1, 127, 1, 3, 5, 15, 17, 51, 85, 255, 1, 7, 73, 511, 1, 3, 11, 31, 33, 93, 341, 1023, 1, 23, 89, 2047, 1, 3, 5, 7, 9, 13, 15, 21, 35, 39, 45, 63, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095, 1, 8191, 1, 3, 43, 127, 129, 381, 5461, 16383

Offset: 1

Seiichi Manyama, Mar 12 2023

			Triangle begins:
  1;
  1,   3;
  1,   7;
  1,   3,  5,   15;
  1,  31;
  1,   3,  7,    9, 21, 63;
  1, 127;
  1,   3,  5,   15, 17, 51,  85,  255;
  1,   7, 73,  511;
  1,   3, 11,   31, 33, 93, 341, 1023;
  1,  23, 89, 2047;

Seiichi Manyama, Rows n = 1..64, flattened
Index entries for sequences related to divisors of numbers

Subsequence of A027750.
Cf. A000225, A049479 (2nd column), A075708 (row sums).
Cf. A374237 (analogous for 2^n + 1).

T:= n-> sort([numtheory[divisors](2^n-1)[]])[]:
seq(T(n), n=1..12);  # Alois P. Heinz, Oct 20 2024

Divisors[2^Range[15] - 1] (* Paolo Xausa, Jul 02 2024 *)

A177710 Values of n where A046801(n) produces a record.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 20, 24, 36, 48, 60, 72, 84, 108, 120, 144, 168, 180, 288, 300, 360, 420, 540, 660, 720, 780, 840, 900, 1080

Offset: 1

J. Lowell, May 11 2010

			240 doesn't qualify because 2^180-1 has 6291456 divisors but 2^240-1 has only 4718592 divisors.

Vjekoslav Kovač and Florian Luca, On the number of divisors of Mersenne numbers, arXiv:2506.04883 [math.NT], 2025. See Table 1 p. 8.

Cf. A046801.

Corrected and extended by Alois P. Heinz, Sep 13 2011

a(24)-a(30) from Amiram Eldar, Sep 02 2019

A059499 a(n) = |{m : multiplicative order of 2 mod m = n}|.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 3, 16, 1, 5, 5, 8, 1, 24, 1, 38, 9, 11, 3, 68, 6, 5, 4, 54, 7, 79, 1, 16, 11, 5, 13, 462, 3, 5, 13, 140, 3, 123, 7, 110, 54, 11, 7, 664, 2, 114, 29, 118, 7, 124, 59, 188, 13, 55, 3, 4456, 1, 5, 82, 96, 5, 353, 3, 118, 11, 485, 7

Offset: 1

Vladeta Jovovic, Feb 04 2001

nonn

Also, number of primitive factors of 2^n - 1 (cf. A212953). - Max Alekseyev, May 03 2022
The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m). See A002326.
a(n) is odd iff n is squarefree, A005117. - Thomas Ordowski, Jan 18 2014
The set S for which a(n) = |S| contains an odd number of prime powers p^k, where k > 0 and p == 3 (mod 4), iff n is squarefree and greater than one. - Isaac Saffold, Dec 28 2019

			a(3) = |{7}| = 1, a(4) = |{5,15}| = 2, a(6) = |{9,21,63}| = 3.

Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 200 terms from Alois P. Heinz)

Column k=2 of A212957.
Primitive factors of b^n - 1: this sequence (b=2), A059885 (b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A001037, A046801, A058943, A059912, A112927, A212953.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(2^d-1), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, May 31 2012

a[n_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, 2^d - 1], {d, Divisors[n]}]; Table[a[n], {n, 1, 71} ] (* Jean-François Alcover, Dec 12 2012 *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(2^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{d|n} A008683(n/d) * A046801(d) = Sum_{d|A007947(n)} A008683(d) * A046801(n/d). - Max Alekseyev, May 03 2022

a(n) = 1 iff 2^n-1 is noncomposite. a(prime(n)) = 2^A088863(n)-1. - Thomas Ordowski, Jan 16 2014

More terms from John W. Layman, Mar 22 2002

More terms from Alois P. Heinz, May 31 2012

A046798 Number of divisors of 2^n + 1.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 4, 4, 2, 8, 6, 4, 4, 4, 8, 12, 2, 4, 16, 4, 4, 12, 8, 4, 8, 16, 16, 20, 4, 8, 48, 4, 4, 24, 16, 32, 16, 8, 16, 12, 4, 8, 64, 4, 8, 64, 32, 8, 8, 8, 64, 48, 8, 8, 64, 48, 8, 24, 8, 16, 16, 4, 32, 64, 4, 64, 64, 8, 12, 24, 96, 8, 32, 8, 32, 96, 16, 64, 768, 4, 8, 192, 32, 64

Offset: 0

Labos Elemer

nonn

a(n) is odd iff n = 3, as a consequence of the Catalan-Mihăilescu theorem. - Bernard Schott, Oct 05 2021

			a(7)=4, because 2^7 + 1 = 129 has 4 divisors.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..500 from T. D. Noe, terms 501..1062 from Amiram Eldar, term 1108 from Tyler Busby)
Wikipedia, Catalan's conjecture.

Cf. A000005, A000051, A002587, A002589, A046801, A054992, A057957, A059886, A274906, A344897.

[NumberOfDivisors(2^n+1): n in [0..100]]; // Vincenzo Librandi, Feb 05 2018

a:= n-> numtheory[tau](2^n+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 23 2021

A046798[n_IntegerQ]:=DivisorSigma[0,1+2^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
DivisorSigma[0, 1 + 2^#] & /@ Range[0, 83] (* Jayanta Basu, Jun 29 2013 *)
Table[DivisorSigma[0, 2^n + 1], {n, 0, 100}] (* Vincenzo Librandi, Feb 05 2018 *)

a(n) = numdiv(2^n+1); \\ Michel Marcus, Mar 18 2017

from sympy.ntheory import divisor_count
def A046798(n): return divisor_count(2**n + 1) # Indranil Ghosh, Mar 18 2017

a(n) = A000005(A000051(n)). - Michel Marcus, Mar 18 2017

A070528 Number of divisors of 10^n-1 (999...999 with n digits).

Original entry on oeis.org

3, 6, 8, 12, 12, 64, 12, 48, 20, 48, 12, 256, 24, 48, 128, 192, 12, 640, 6, 384, 256, 288, 6, 2048, 96, 192, 96, 768, 96, 16384, 24, 6144, 128, 192, 384, 5120, 24, 24, 128, 6144, 48, 49152, 48, 4608, 1280, 192, 12, 16384, 48, 3072, 512, 1536, 48, 12288, 768

Offset: 1

Henry Bottomley, May 02 2002

			a(7)=12 since the divisors of 9999999 are 1, 3, 9, 239, 717, 2151, 4649, 13947, 41841, 1111111, 3333333, 9999999.

Table of n, a(n) for n = 1..352

Cf. A004023 (indices of 6's).
Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895.
Cf. A046801, A005422, A102347, A046053, A057951, A007138, A003060, A059892, A085035.

DivisorSigma[0,#]&/@(10^Range[60]-1) (* Harvey P. Dale, Jan 14 2011 *)
Table[DivisorSigma[0, 10^n - 1], {n, 60}] (* T. D. Noe, Aug 18 2011 *)

a(n) = numdiv(10^n - 1); \\ Michel Marcus, Sep 08 2015

a(n) = A000005(A002283(n)).
a(n) = Sum_{d|n} A059892(d).
a(n) = A070529(n)*(A007949(n)+3)/(A007949(n)+1).

Terms to a(280) in b-file from Hans Havermann, Aug 19 2011
a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) in b-file from Max Alekseyev, May 04 2022

A366709 Number of divisors of 12^n-1.

Original entry on oeis.org

2, 4, 4, 16, 4, 32, 8, 64, 16, 16, 12, 256, 8, 64, 64, 512, 8, 512, 4, 192, 32, 48, 16, 4096, 16, 192, 64, 1024, 32, 8192, 32, 2048, 192, 64, 512, 16384, 8, 64, 128, 12288, 16, 12288, 32, 3072, 4096, 256, 8, 262144, 32, 1024, 64, 6144, 128, 65536, 192, 8192

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

			a(3)=4 because 12^3-1 has divisors {1, 11, 157, 1727}.

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A000005, A024140, A046801, A070528, A366707, A366708, A366710, A366683.

a:=n->numtheory[tau](12^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 12^Range[100]-1]
```
PARI
```
a(n) = numdiv(12^n-1);
```

a(n) = sigma0(12^n-1) = A000005(A024140(n)).

A366683 Number of divisors of 11^n-1.

Original entry on oeis.org

4, 16, 16, 40, 12, 192, 16, 96, 32, 96, 16, 1920, 16, 128, 96, 448, 8, 1024, 8, 480, 768, 1024, 32, 18432, 128, 512, 64, 2560, 16, 9216, 32, 2048, 512, 256, 192, 20480, 64, 512, 4096, 4608, 512, 36864, 16, 10240, 384, 2048, 32, 1376256, 128, 4096, 512, 2560

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

			a(3)=16 because 11^3-1 has divisors {1, 2, 5, 7, 10, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330}.

Max Alekseyev, Table of n, a(n) for n = 1..316

Cf. A024127, A000005, A366681, A366682, A366684, A366685.

Cf. A046801, A070528, A366709.

a:=n->numtheory[tau](11^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 11^Range[100]-1]
```
PARI
```
a(n) = numdiv(11^n-1);
```

a(n) = sigma0(11^n-1) = A000005(A024127(n)).

A246600 Number of divisors d of n with property that the binary representation of d can be obtained from the binary representation of n by changing any number of 1's to 0's.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 4, 3, 2, 2, 2, 2, 4, 2, 2, 6, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 4, 2, 3, 2, 2, 4, 2

Offset: 1

N. J. A. Sloane, Sep 06 2014

Equivalently, the number of divisors d of n such that the bitwise OR of n and d is equal to n. - Chai Wah Wu, Sep 06 2014
Equivalently, the number of divisors d of n such that the bitwise AND of n and d is equal to d. - Amiram Eldar, Dec 15 2022
The sums of the first 10^k terms for k = 1, 2, ..., are 16, 224, 2580, 26920, 273407, 2745100, 27440305, 274127749, 2738936912, 27373288534, 273631055291, 2735755647065, ... . Conjecture: The asymptotic mean of this sequence is 1 + Sum_{k>=1} 1/(k*2^A000120(k)) = 2.7351180693... . - Amiram Eldar, Apr 07 2023

			12 = 1100_2; only the divisors 4 = 0100_2 and 12 = 1100_2 satisfy the condition, so(12)=2.
15 = 1111_2; all divisors 1,3,5,15 satisfy the condition, so a(15)=4.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A000005, A000120, A046801, A246601, A047999.

A246600:=proc(n)
    local a, d, s, t, i, sw;
    a:=0;
    s:=convert(n, base, 2);
    for d in numtheory[divisors](n) do
        sw:= false;
        t:=convert(d, base, 2);
        for i from 1 to nops(t) do
            if t[i]>s[i] then
                sw:= true;
            fi;
        od:
        if not sw then
            a:=a+1;
        fi;
    od;
    a;
end;
seq(A246600(n), n=1..100);

a[n_] := DivisorSum[n, Boole[BitOr[#, n] == n]&]; Array[a, 100] (* Jean-François Alcover, Dec 02 2015, adapted from PARI *)

a(n)=sumdiv(n,d,bitor(d,n)==n) \\ Charles R Greathouse IV, Sep 29 2014

from sympy import divisors
def A246600(n):
    return sum(1 for d in divisors(n) if n|d == n)
# Chai Wah Wu, Sep 06 2014

a(2^i) = 1.
a(odd prime) = 2.
a(n) <= 2^wt(n)-1, where wt(n) = A000120(n).
a(n) = Sum_{d|n} A047999(n,d), where A047999(n,d) = binomial(n,d) mod 2. - Ridouane Oudra, May 03 2019
From Amiram Eldar, Dec 15 2022: (Start)
a(2*n) = a(n), and therefore a(m*2^k) = a(m) for m odd and k>=0.
a(2^n-1) = A000005(2^n-1) = A046801(n). (End)

A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A046051, the number of prime factors of Mersenne number 2^n-1.

The number of prime factors in the primitive part of 2^n-1. - T. D. Noe, Jul 19 2008

			a(11) = 2 because cyclotomic(11,2) = 2047, which has two factors: 23 and 89.

Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 500 term from T. D. Noe)
Vjekoslav Kovač and Florian Luca, On the number of divisors of Mersenne numbers, arXiv:2506.04883 [math.NT], 2025. See Table 2 p. 9.
H. Mishima, Factorization of Cyclotomic Numbers
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Eric Weisstein's World of Mathematics, Möbius Transform

omega(Phi(n,x)): this sequence (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A005420, A019320, A046051, A046801, A059499, A064078, A112927, A212953.

Join[{0}, Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 2]]][[2]], {n, 2, 100}]]

a(n) = #factor(polcyclo(n, 2))~; \\ Michel Marcus, Mar 06 2015

A366621 Number of divisors of 6^n-1.

Original entry on oeis.org

2, 4, 4, 8, 6, 16, 4, 16, 16, 48, 8, 128, 8, 48, 48, 64, 32, 128, 8, 384, 16, 32, 32, 512, 32, 128, 64, 384, 4, 1536, 8, 512, 64, 256, 96, 8192, 64, 64, 64, 3072, 8, 768, 32, 512, 1536, 256, 16, 8192, 32, 512, 512, 2048, 16, 2048, 96, 12288, 128, 64, 16

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(4)=8 because 6^4-1 has divisors {1, 5, 7, 35, 37, 185, 259, 1295}.

Max Alekseyev, Table of n, a(n) for n = 1..430

Cf. A024062, A000005, A057955, A059888, A274907, A366613, A366620, A366622, A366623.

Cf. A046801, A366575, A366602, A366612, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](6^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 6^Range[100]-1]
```
PARI
```
a(n) = numdiv(6^n-1);
```

a(n) = sigma0(6^n-1) = A000005(A024062(n)).

cp's OEIS Frontend

A361438 Triangle T(n,k), n >= 1, 1 <= k <= A046801(n), read by rows, where T(n,k) is k-th smallest divisor of 2^n-1.

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Maple

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A177710 Values of n where A046801(n) produces a record.

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A059499 a(n) = |{m : multiplicative order of 2 mod m = n}|.

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A046798 Number of divisors of 2^n + 1.

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Magma

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A070528 Number of divisors of 10^n-1 (999...999 with n digits).

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Mathematica

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A366709 Number of divisors of 12^n-1.

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Maple

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A366683 Number of divisors of 11^n-1.

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