A084006 -id:A084006 - cp's OEIS frontend

A084007 a(n) = A084006(n)^(1/2).

6, 9, 33, 66, 99, 333, 666, 999, 3333, 6666, 9999, 33333, 66666, 99999, 333333, 666666, 999999, 3333333, 6666666, 9999999, 33333333, 66666666, 99999999, 333333333, 444444444, 555555555, 666666666, 777777777, 888888888, 999999999

Offset: 0

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

Cf. A084004, A084005, A084006.

from itertools import count, islice
from math import prod, isqrt
from sympy import factorint
def A084007_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)
        y = isqrt(x*m)
        yield from (j*y for j in range(isqrt(10**(l-1)//x)+1,isqrt(m//x)+1))
A084007_list = list(islice(A084007_gen(),30)) # Chai Wah Wu, Mar 20 2025

Pattern exhibited by early terms does not continue without interruption. First disruption occurs at a(25)=444444444. Terms with k-digits exhibit the earlier pattern where (10^k-1)/9 is squarefree and k=9 is the first occurrence where (10^k-1)/9 is not squarefree. Others occur at k=18, 22, 27, 36, 42, 44, 45. - Ray Chandler, Aug 04 2003

More terms from Ray Chandler, May 31 2003

More terms from Ray Chandler, Aug 04 2003

A046412 Lengths of nonsquarefree repunits.

Original entry on oeis.org

9, 18, 22, 27, 36, 42, 44, 45, 54, 63, 66, 72, 78, 81, 84, 88, 90, 99, 108, 110, 111, 117, 126, 132, 135, 144, 153, 154, 156, 162, 168, 171, 176, 180, 189, 198, 205, 207, 210, 216, 220, 222, 225, 234, 242, 243, 252, 261, 264, 270, 272, 279, 286, 288, 294, 297, 306, 308, 312, 315, 324, 330, 333, 336, 342, 351, 352

Offset: 1

Patrick De Geest, Jul 15 1998

This is the set of all positive multiples of all positive members of A087094. What is the asymptotic density of this set? - Jeppe Stig Nielsen, Dec 28 2015

Patrick De Geest, Repunits prime factors
Shyam Sunder Gupta, Repunit Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 11, 327-352.
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A004022, A046053, A084006, A087094.

remove(t -> numtheory:-issqrfree((10^t-1)/9), [$1..90]); # Robert Israel, Dec 30 2015

Select[Range[300],!SquareFreeQ[(10^#-1)/9]&] (* Harvey P. Dale, Jun 26 2013 *)

isok(n) = ! issquarefree((10^n-1)/9); \\ Michel Marcus, Dec 31 2015

a(n)=k where (10^k-1)/9 is not squarefree. - Ray Chandler, Aug 10 2003

Terms to a(60) from Ray Chandler, Aug 10 2003

a(61)-a(67) from Max Alekseyev, Apr 29 2022

A084004 Squares obtained as a concatenation of k and 10's complement of k.

Original entry on oeis.org

64, 7921, 9604, 164836, 351649, 996004, 19758025, 20647936, 29757025, 30846916, 97990201, 99960004, 1203187969, 1975180249, 3086469136, 4265657344, 8143618564, 9999600004, 115644884356, 131504868496, 132231867769, 140039859961, 173879826121, 339900660100, 391600608400

Offset: 1

Amarnath Murthy, May 23 2003

Andrew Howroyd, Table of n, a(n) for n = 1..65

Cf. A084005, A084006, A084007.

b(n)={my(k=logint(n,10)+1); (n+1)*10^k - n}
{ for(k=1, 10^6, my(x=b(k)); if(issquare(x), print1(x, ", "))) } \\ Andrew Howroyd, Sep 22 2024

a(n) = A084005(n)^2. - Andrew Howroyd, Sep 21 2024

More terms from Ray Chandler, May 31 2003

Offset changed and a(22) onwards from Andrew Howroyd, Sep 21 2024

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

A084005 a(n) = sqrt(A084004(n)).

Original entry on oeis.org

8, 89, 98, 406, 593, 998, 4445, 4544, 5455, 5554, 9899, 9998, 34687, 44443, 55556, 65312, 90242, 99998, 340066, 362636, 363637, 374219, 416989, 583010, 625780, 636362, 637363, 659933, 702703, 703702, 713285, 714286, 780625, 922076, 923077

Offset: 1

Amarnath Murthy, May 23 2003

Cf. A084004, A084006, A084007.

More terms from Ray Chandler, May 31 2003

Offset changed by Andrew Howroyd, Sep 22 2024

cp's OEIS Frontend

A084007 a(n) = A084006(n)^(1/2).

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A046412 Lengths of nonsquarefree repunits.

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A084004 Squares obtained as a concatenation of k and 10'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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A084005 a(n) = sqrt(A084004(n)).

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