A070529 -id:A070529 - cp's OEIS frontend

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

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

A102146 a(n) = sigma(10^n - 1), where sigma(n) is the sum of positive divisors of n.

Original entry on oeis.org

13, 156, 1520, 15912, 148512, 2042880, 14508000, 162493344, 1534205464, 16203253248, 144451398000, 2063316971520, 14903272088640, 158269280832000, 1614847741624320, 17205180696931968, 144444514193267496

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (terms 1..322 from Ray Chandler)
C. Caldwell, Sigma function.

Cf. A000203, A001270, A002283, A003020, A005422, A046053, A046107, A046412, A046415, A046416, A046417, A046418, A046419, A046420, A057951, A059892, A061075, A070528, A070529, A081317, A081318, A085035, A095370, A095413, A095414, A095417, A095418, A102347, A102380, A112505, A147556, A295503, A366669.

DivisorSigma[1,10^Range[20]-1] (* Harvey P. Dale, Jan 05 2012 *)

a(n) = sigma(10^n-1); \\ Michel Marcus, Apr 22 2017

a(n) = A000203(A002283(n)). - Ray Chandler, Apr 22 2017

A109492 Divisors of 111111.

Original entry on oeis.org

1, 3, 7, 11, 13, 21, 33, 37, 39, 77, 91, 111, 143, 231, 259, 273, 407, 429, 481, 777, 1001, 1221, 1443, 2849, 3003, 3367, 5291, 8547, 10101, 15873, 37037, 111111

Offset: 1

Philippe Deléham, Aug 28 2005

Note that the smaller repunits R3=111, R4=1111, R5=11111 are semiprimes and have only 4 divisors, which is again the case for the next repunit R7=1111111. - M. F. Hasler, Oct 13 2011

Eric Weisstein's World of Mathematics, Repunit
Index entries for sequences related to divisors of numbers

Cf. A070529, A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895. [M. F. Hasler, Oct 13 2011]

Cf. A199799 (totatives of 111111), A154549 (111111*n).

Divisors[111111] (* Paolo Xausa, Sep 27 2023 *)

divisors(111111) \\ Michel Marcus, Sep 08 2015

A103759 a(n) = sigma((10^n - 1)/9), where sigma(n) is the sum of positive divisors of n.

Original entry on oeis.org

1, 12, 152, 1224, 11424, 204288, 1116000, 12499488, 164831992, 1246404096, 11111646000, 206331697152, 1146405545280, 12174560064000, 161484774162432, 1323475438225536, 11111116476405192, 232965361825996800

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Mar 28 2005

nonn

Ray Chandler, Table of n, a(n) for n = 1..322

Cf. A000203, A002275, A070529.

Table[DivisorSigma[1,FromDigits[PadRight[{},n,1]]],{n,20}] (* Harvey P. Dale, Jul 13 2022 *)

a(n) = sigma((10^n - 1)/9); \\ Michel Marcus, Sep 08 2015

a(n) = A000203(A002275(n)). - Michel Marcus, Sep 08 2015

A197308 Divisors of 11111111.

Original entry on oeis.org

1, 11, 73, 101, 137, 803, 1111, 1507, 7373, 10001, 13837, 81103, 110011, 152207, 1010101, 11111111

Offset: 1

M. F. Hasler, Oct 13 2011

Index entries for sequences related to divisors of numbers.

Cf. A109492, A197309, A070529, A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895.

Divisors[11111111] (* Harvey P. Dale, Nov 12 2021 *)

PARI
```
divisors(11111111)
```

A197309 Divisors of the 9th repunit 111111111.

Original entry on oeis.org

1, 3, 9, 37, 111, 333, 333667, 1001001, 3003003, 12345679, 37037037, 111111111

Offset: 1

M. F. Hasler, Oct 13 2011

The prime factorization of (10^9 - 1)/9 is 3^2 * 37 * 333667. - Alonso del Arte, Apr 27 2014

Index entries for sequences related to divisors of numbers.

Cf. A109492, A197308, A070529, A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895.

Divisors[(10^9 - 1)/9] (* Alonso del Arte, Apr 27 2014 *)

divisors(10^9\9) \\ Charles R Greathouse IV, Jun 21 2017

A197318 Divisors of the repunit 111111111111 = A002275(12).

Original entry on oeis.org

1, 3, 7, 11, 13, 21, 33, 37, 39, 77, 91, 101, 111, 143, 231, 259, 273, 303, 407, 429, 481, 707, 777, 1001, 1111, 1221, 1313, 1443, 2121, 2849, 3003, 3333, 3367, 3737, 3939, 5291, 7777, 8547, 9191, 9901, 10101, 11211, 14443, 15873, 23331, 26159, 27573, 29703

Offset: 1

M. F. Hasler, Oct 13 2011

The sequence is marked "full" since even though they don't fit into the three lines above, all 128 terms are known and available in the b-file or using the given PARI code.

M. F. Hasler, Table of n, a(n) for n = 1..128 (complete sequence)
Index entries for sequences related to divisors of numbers.

Cf. A109492, A197308, A197309, A070529, A018282, A018766, A027890-A027895, A027889.

Divisors[111111111111] (* Paolo Xausa, Jul 04 2024 *)

PARI
```
divisors(1e12\9)
```

A227815 Composite numbers n divisible by their concatenated exponents in prime factorization.

Original entry on oeis.org

4, 16, 22, 27, 33, 55, 63, 77, 143, 187, 209, 222, 248, 253, 256, 319, 341, 407, 451, 473, 484, 517, 555, 583, 649, 656, 671, 737, 777, 781, 803, 837, 869, 913, 979, 1067, 1111, 1133, 1152, 1177, 1199, 1221, 1243, 1397, 1441, 1443, 1507, 1529, 1639, 1661, 1727

Offset: 1

Michel Lagneau, Jul 31 2013

The numbers 2^(2^m), m = 1, 2,... are in the sequence. A majority of semiprimes of the form 11*p where is p prime different from 11 are in the sequence. The numbers of the form p*111 = p*3*37 where p is prime different from 3 or 37 are in the sequence. In the general case, the numbers of the form n = p_1*p_2*...*p_m*R where the p_k are prime numbers and R is a repunit number (A002275) with q digits "1" and (q-m) prime divisors are in the sequence, for example the numbers of the form n = p_1*p_2*p_3*p_4*11111111 = p_1*p_2*p_3*p_4*11*73*101*137 are in the sequence if the primes p_k are different from 11, 73, 101 or 137. So, 11111111 divides n.

			248 = 2^3*31 => 31 is the concatenate exponents 3 and 1, so 31 divides 248.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002275, A037916, A070529.

with(numtheory):for n from 1 to 10000 do:x:=ifactors(n):y:=x[2];n1:=nops(y):s:=0:for i from 1 to n1 do:z:=y[i][2]:s:=s+z*10^(n1-i):od:if type(n,prime)=false and irem(n,s)=0 then printf(`%d, `, n):else fi:od:

With[{predicate = And[CompositeQ[#], Divisible[#, FromDigits[Join @@ IntegerDigits@(Last /@ FactorInteger[#])]]] &},
Select[Range[10000], predicate]] (* Sidney Cadot, Feb 19 2023 *)

from sympy import isprime, factorint
def ok(n): return n > 1 and not isprime(n) and n%int("".join(str(e) for e in factorint(n).values())) == 0
print([k for k in range(1728) if ok(k)]) # Michael S. Branicky, Feb 19 2023

cp's OEIS Frontend

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

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A102146 a(n) = sigma(10^n - 1), where sigma(n) is the sum of positive divisors of n.

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A109492 Divisors of 111111.

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A103759 a(n) = sigma((10^n - 1)/9), where sigma(n) is the sum of positive divisors of n.

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A197308 Divisors of 11111111.

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A197309 Divisors of the 9th repunit 111111111.

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A197318 Divisors of the repunit 111111111111 = A002275(12).

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A227815 Composite numbers n divisible by their concatenated exponents in prime factorization.

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