A352992 - OEIS

A352992

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

%I #61 Apr 18 2022 22:46:23

%S 1,3,53,1753,753,60753,660753,9660753,99660753,899660753,3899660753,

%T 33899660753,233899660753,7233899660753,97233899660753,

%U 497233899660753,1497233899660753,31497233899660753,631497233899660753,9631497233899660753,59631497233899660753,559631497233899660753

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

%C 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).

%C 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.

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

%C 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.

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

%e a(1) = 3 because 3^3 = 27;

%e a(2) = 53 because 53^2 = 148877;

%e a(3) = 1753 because 1753^3 = 5386984777;

%e a(4) = 753 because 753^2 = 426957777;

%e a(5) = 60753 because 60753^3 = 224234888577777.

%e Table with a(n) and A225401(n-1)

%e ---------------------------------------------------------------------------

%e | | a(n) | a'(n) | A225401(n-1) | concatenation |

%e | n | with "exactly" | without "exactly" | = b(n-1) | b(n-1)...b(0) |

%e ---------------------------------------------------------------------------

%e 0 1 1

%e 1 3 3 3 ...3

%e 2 53 53 5 ...53

%e 3 1753 753 7 ...753

%e 4 753 753 0 ...0753

%e 5 60753 60753 6 ...60753

%e 6 660753 660753 6 ...660753

%e 7 9660753 9660753 9 ...9660753

%e ..........................................................................

%e Also, as A225401(23) = 0, we have from a(21) up to a(25):

%e a(21) = 559631497233899660753;

%e a(22) = 3559631497233899660753;

%e a(23) = 193559631497233899660753, found by _Marius A. Burtea_;

%e a(24) = 93559631497233899660753;

%e a(25) = 2093559631497233899660753.

%o (Python)

%o def a(n):

%o k, s, target = 1, "1", "7"*n

%o while s.rstrip("7") + target != s: k += 1; s = str(k**3)

%o return k

%o print([a(n) for n in range(8)]) # _Michael S. Branicky_, Apr 14 2022

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

%K nonn,base

%O 0,2

%A _Bernard Schott_, Apr 14 2022

%E a(8)-a(9) from _Marius A. Burtea_, Apr 14 2022