A109492 -id:A109492 - cp's OEIS frontend

A083230 Number of repunit divisors of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Jun 01 2003

Differs from A043284 (maximal run length in decimal expansion) from a(100) on. - M. F. Hasler, Oct 18 2019

			n = 110, divisors are {1, 2, 5, 10, 11, 22, 55, 110} with two repunits: 1 and 11, therefore a(110) = 2.
n = 111, divisors are {1, 3, 37, 111} with two repunits: 1 and 111, therefore a(111) = 2.
n = 111111, divisors are {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} with four repunits: 1, 11, 111 and 111111, therefore a(111111) = 4.

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1221 from Andrew Howroyd)

Cf. A000005, A002275, A043284, A065444.

Cf. A109492.

A083230[n_]:=Count[IntegerDigits[Divisors[n]],{1..}];Array[A083230,100] (* Paolo Xausa, Sep 27 2023 *)

a(n)={my(s=0, k=1); while(k<=n, if(n%k==0, s++); k=10*k+1); s} \\ Andrew Howroyd, Aug 07 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A065444. - Amiram Eldar, Apr 17 2025

A109489 Value of Product[k/sd(k,2),k=1..n], where sd(k,b) is the sum of the digits of k represented in base b.

Original entry on oeis.org

1, 2, 3, 12, 30, 90, 210, 1680, 7560, 37800, 138600, 831600, 3603600, 16816800, 63063000, 1009008000, 8576568000, 77189112000, 488864376000, 4888643760000, 34220506320000, 250950379680000, 1442964683160000, 17315576197920000

Offset: 1

John W. Layman, Jun 29 2005

It appears that Product[k/sd(k,b),k=1..n] is an integer for all integers n>0 and b>1. Is this known or easy to prove?

It is not true! The product is not an integer for b=2 and n=422 (it has a denominator of 5). B-file contains all terms before that. - Robert Israel, Jan 21 2018

			The base 2 representations of 1,2,3,4 are 1,10,11,100 so a(4)=(1/1)(2/1)(3/2)(4/1)=12.

Robert Israel, Table of n, a(n) for n = 1..421

Cf. A003601, A109490, A109491, A109492.

P:= 1: A[1]:= P:
for n from 2 to 100 do
  P:= P*n/convert(convert(n,base,2),`+`);
  A[n]:= P;
od:
seq(A[i],i=1..100); # Robert Israel, Jan 21 2018

a(n) = prod(k=1, n, k/hammingweight(k)); \\ Michel Marcus, Jul 10 2014

A177769 a(n) = 111*n.

Original entry on oeis.org

111, 222, 333, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443, 1554, 1665, 1776, 1887, 1998, 2109, 2220, 2331, 2442, 2553, 2664, 2775, 2886, 2997, 3108, 3219, 3330, 3441, 3552, 3663, 3774, 3885, 3996, 4107, 4218, 4329, 4440, 4551, 4662, 4773, 4884, 4995, 5106

Offset: 1

Paul Curtz, May 13 2010

The reference contains also sequences A102807, A109344, A075415, and A109492.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Émile Fourrey, Le nombre 37, Récreations arithmétiques, 1899 and later, Vuibert, Paris, page 11.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A075415, A102807, A109344, A109492.

[111*n: n in [1..50]]; // Vincenzo Librandi, Aug 04 2011

G.f.: 111*x/(x-1)^2.
a(n) = 2*a(n-1) - a(n-2).
a(n) = a(n-1) + 111.
E.g.f.: 111*x*exp(x). - Stefano Spezia, Sep 15 2023

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

A199799 Totatives of 111111.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 17, 19, 20, 23, 25, 29, 31, 32, 34, 38, 40, 41, 43, 46, 47, 50, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 76, 79, 80, 82, 83, 85, 86, 89, 92, 94, 95, 97, 100, 101, 103, 106, 107, 109, 113, 115, 116, 118, 122, 124, 125, 127, 128, 131, 134

Offset: 1

Reinhard Zumkeller, Nov 11 2011

a(n) and 111111 are coprime, 111111 = 3*7*11*13*37; empty intersections with A008585, A008593, A008595, or A085959; sequence is finite with 51840 terms, A000010(111111) = 51840, last term: a(51840) = 111110.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Totative

Cf. A109492 (divisors of 111111).

a199799 n = a199799_list !! (n-1)
a199799_list = [x | x <- [1..111111], gcd x 111111 == 1]

Select[Range[200],CoprimeQ[#,111111]&] (* Paolo Xausa, Sep 27 2023 *)

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)
```

cp's OEIS Frontend

A083230 Number of repunit divisors of n.

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A109489 Value of Product[k/sd(k,2),k=1..n], where sd(k,b) is the sum of the digits of k represented in base b.

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Maple

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A177769 a(n) = 111*n.

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Magma

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A197308 Divisors of 11111111.

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A197309 Divisors of the 9th repunit 111111111.

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A199799 Totatives of 111111.

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Haskell

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

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