A046412 -id:A046412 - cp's OEIS frontend

A070529 Number of divisors of repunit 111...111 (with n digits).

1, 2, 4, 4, 4, 32, 4, 16, 12, 16, 4, 128, 8, 16, 64, 64, 4, 384, 2, 128, 128, 96, 2, 1024, 32, 64, 64, 256, 32, 8192, 8, 2048, 64, 64, 128, 3072, 8, 8, 64, 2048, 16, 24576, 16, 1536, 768, 64, 4, 8192, 16, 1024, 256, 512, 16, 8192, 256, 4096

Offset: 1

Henry Bottomley, May 02 2002

			a(9) = 12 since the divisors of 111111111 are 1, 3, 9, 37, 111, 333, 333667, 1001001, 3003003, 12345679, 37037037, 111111111.

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

Cf. A000005, A002275, A070528, A051064, A004023, A046413, A003020, A095370, A176973, A046053, A046412, A102380.

a(n) = numdiv(10^n\9); \\ Michel Marcus, Sep 09 2015

a(n) = A000005(A002275(n)).
a(n) = A070528(n)*A051064(n)/(A051064(n)+2).
a(A004023(n)) = 2. - Michel Marcus, Sep 09 2015
a(A046413(n)) = 4. - Bruno Berselli, Sep 09 2015

Terms to a(280) in b-file from Hans Havermann, Aug 20 2011
a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) ib 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

A084006 Squares arising as a concatenation of k and 9's complement of k.

Original entry on oeis.org

36, 81, 1089, 4356, 9801, 110889, 443556, 998001, 11108889, 44435556, 99980001, 1111088889, 4444355556, 9999800001, 111110888889, 444443555556, 999998000001, 11111108888889, 44444435555556, 99999980000001

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 23 2003

From Robert Israel, Sep 09 2020: (Start)
Numbers of the form j^2*x*(10^k-1) where x = A007913(10^k-1) and 10^(k-1)+1 <= j^2*x <= 10^k-1.
If k >= 2 is not in A046412, there are only three terms with 2*k digits, namely (10^k-1)^2/9, 4*(10^k-1)^2/9, and 9*(10^k-1)^2/9.
The first term not of one of those three forms is a(25)=197530863802469136.
(End)

			1089 = 33^2 is a concatenation of 10 and 89, 10+89 = 99.

Robert Israel, Table of n, a(n) for n = 1..367

Cf. A007913, A046412, A084004, A084005, A084007.

f:= proc(k) local F,x,p,t;
  p:= 10^k-1;
  F:= select(t -> t[2]::odd, ifactors(p)[2]);
  x:= mul(t[1],t=F);
  seq(j^2*x*p, j=ceil(sqrt((10^(k-1)+1)/x))..floor(sqrt(p/x)))
end proc:
map(f, [$1..20]); # Robert Israel, Sep 09 2020

from itertools import count, islice
from math import prod, isqrt
from sympy import factorint
def A084006_gen(): # generator of terms
    for l in count(1):
        m = 10**l-1
        x = prod(p for p, e in factorint(m).items() if e&1)
        yield from (j**2*x*m for j in range(isqrt(10**(l-1)//x)+1,isqrt(m//x)+1))
A084006_list = list(islice(A084006_gen(),20)) # Chai Wah Wu, Mar 20 2025

More terms from Ray Chandler, May 31 2003

Offset changed by Robert Israel, Sep 09 2020

A087094 a(n) = smallest k such that (10^k-1)/9 == 0 mod prime(n)^2, or 0 if no such k exists.

Original entry on oeis.org

0, 9, 0, 42, 22, 78, 272, 342, 506, 812, 465, 111, 205, 903, 2162, 689, 3422, 3660, 2211, 2485, 584, 1027, 3403, 3916, 9312, 404, 3502, 5671, 11772, 12656, 5334, 17030, 1096, 6394, 22052, 11325, 12246, 13203, 27722, 7439, 31862, 32580, 18145, 37056, 19306

Offset: 1

Ray Chandler, Aug 10 2003

nonn

For a given a(n)>0, all of the values of k such that (10^k-1)/9=0 mod prime(n)^2 is given by the sequence a(n)*A000027, i.e. integral multiples of a(n). For example, for n=2, prime(2)=3, a(n)=9, the set of values of k for which (10^k-1)/9=0 mod 3^2 is 9*A000027=9,18,27,36,45,...
The union of the collection of sequences formed from the nonzero terms of a(n)*A000027, gives the values of k for which (10^k-1)/9 is not squarefree, see A046412. All of terms of the sequence a(n) are integer multiples of prime(n) for primes <1000 except for a(93)=486 where prime(93)=487. Conjecture: there are no 0 terms after a(3).
That conjecture is easily proved, for a(n) is just the multiplicative order of 10 modulo (prime(n))^2 for n>3. - Jeppe Stig Nielsen, Dec 28 2015

			a(2)=9 since 9 is least value of k for which (10^k-1)/9=0 mod 3^2.

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

Cf. A000040, A000042, A046412, A084006, A084007.

0,9,0,seq(numtheory:-order(10,ithprime(i)^2), i=4..100); # Robert Israel, Dec 30 2015

a(n)=p=prime(n);10%p==0 && return(0);for(k=1,p^2,((10^k-1)/9) % p^2 == 0 && return(k));error() \\ Jeppe Stig Nielsen, Dec 28 2015

a(n)=p=prime(n);if(10%p==0, 0, 10%p==1, 9, znorder(Mod(10,p^2))) \\ Jeppe Stig Nielsen, Dec 28 2015

For n>3, a(n) = A084680(prime(n)^2) = A084680(A001248(n)), Jeppe Stig Nielsen, Dec 28 2015

A068538 Nonsquarefree repunits.

Original entry on oeis.org

111111111, 111111111111111111, 1111111111111111111111, 111111111111111111111111111, 111111111111111111111111111111111111, 111111111111111111111111111111111111111111, 11111111111111111111111111111111111111111111, 111111111111111111111111111111111111111111111

Offset: 1

Benoit Cloitre, Mar 22 2002

nonn

The number of digits of these repunits form the sequence A046412. - Michel Marcus, Nov 24 2013

lista(nn) = {for (n=1, nn, if (! issquarefree(r = sum(i=0, n, 10^i)), print1(r, ", ")););} \\ Michel Marcus, Nov 24 2013

a(n) = (10^A046412(n) - 1) / 9. - Max Alekseyev, Apr 29 2022

cp's OEIS Frontend

A070529 Number of divisors of repunit 111...111 (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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A084006 Squares arising as a concatenation of k and 9's complement of k.

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A087094 a(n) = smallest k such that (10^k-1)/9 == 0 mod prime(n)^2, or 0 if no such k exists.

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A068538 Nonsquarefree repunits.

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