A084007 -id:A084007 - cp's OEIS frontend

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

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

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

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

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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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