A352992 Smallest positive integer whose cube ends with exactly n 7's.

1, 3, 53, 1753, 753, 60753, 660753, 9660753, 99660753, 899660753, 3899660753, 33899660753, 233899660753, 7233899660753, 97233899660753, 497233899660753, 1497233899660753, 31497233899660753, 631497233899660753, 9631497233899660753, 59631497233899660753, 559631497233899660753

Offset: 0

Author

Bernard Schott, Apr 14 2022

Keywords

• nonn
• base

Comments

When A225401(k) = 0, i.e. k is a term of A353003, then a(k) > a(k+1); 1st example is for k = 3 with a(3) = 1753 > a(4) = 753; otherwise, a(n) < a(n+1).

When n <> k, a(n) coincides with the 'backward concatenation' of A225401(n-1) up to A225401(0), where A225401 is the 10-adic integer x such that x^3 = -7/9 (see table in Example section); when n= k, a(k) must be calculated directly with the definition.

Without "exactly" in the name, terms a'(n) should be: 1, 3, 53, 753, 753, 60753, 660753, ...

There are similar sequences when cubes end with 1, 3, 8 or 9; but there's no similar sequence for squares, because when a square ends in more than three identical digits, these digits are necessarily 0.

Examples

			a(1) = 3 because 3^3 = 27;
a(2) = 53 because 53^2 = 148877;
a(3) = 1753 because 1753^3 = 5386984777;
a(4) = 753 because 753^2 = 426957777;
a(5) = 60753 because 60753^3 = 224234888577777.
Table with a(n) and A225401(n-1)
   ---------------------------------------------------------------------------
   |    |     a(n)       |      a'(n)        | A225401(n-1) |  concatenation |
   | n  | with "exactly" | without "exactly" |  = b(n-1)    |  b(n-1)...b(0) |
   ---------------------------------------------------------------------------
     0           1                 1
     1           3                 3               3                  ...3
     2          53                53               5                 ...53
     3        1753               753               7                ...753
     4         753               753               0               ...0753
     5       60753             60753               6              ...60753
     6      660753            660753               6             ...660753
     7     9660753           9660753               9            ...9660753
  ..........................................................................
Also, as A225401(23) = 0, we have from a(21) up to a(25):
a(21) =     559631497233899660753;
a(22) =    3559631497233899660753;
a(23) =  193559631497233899660753, found by _Marius A. Burtea_;
a(24) =   93559631497233899660753;
a(25) = 2093559631497233899660753.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..996

Crossrefs

Cf. A000578, A017307, A039685, A225401, A353003.

Programs

Maple

f:= proc(n) local t,x;
       t:= 7/9*(10^n-1);
       x:= rhs(op(msolve(x^3=t,10^n)));
       while x^3 mod 10^(n+1) = 10*t+7 do x:= x + 10^n od;
       x
end proc:
f(0):= 1:
map(f, [$0..30]); # Robert Israel, Jul 29 2025

Python

def a(n):
    k, s, target = 1, "1", "7"*n
    while s.rstrip("7") + target != s: k += 1; s = str(k**3)
    return k
print([a(n) for n in range(8)]) # Michael S. Branicky, Apr 14 2022

Formula

When n is not in A353003, a(n) = Sum_{k=0..n-1} A225401(k) * 10^k.

Extensions

a(8)-a(9) from Marius A. Burtea, Apr 14 2022