cp's OEIS Frontend

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

%I A352992 #65 Jul 29 2025 14:32:51
%S A352992 1,3,53,1753,753,60753,660753,9660753,99660753,899660753,3899660753,
%T A352992 33899660753,233899660753,7233899660753,97233899660753,
%U A352992 497233899660753,1497233899660753,31497233899660753,631497233899660753,9631497233899660753,59631497233899660753,559631497233899660753
%N A352992 Smallest positive integer whose cube ends with exactly n 7's.
%C A352992 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 A352992 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 A352992 Without "exactly" in the name, terms a'(n) should be: 1, 3, 53, 753, 753, 60753, 660753, ...
%C A352992 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.
%H A352992 Robert Israel, <a href="/A352992/b352992.txt">Table of n, a(n) for n = 0..996</a>
%F A352992 When n is not in A353003, a(n) = Sum_{k=0..n-1} A225401(k) * 10^k.
%e A352992 a(1) = 3 because 3^3 = 27;
%e A352992 a(2) = 53 because 53^2 = 148877;
%e A352992 a(3) = 1753 because 1753^3 = 5386984777;
%e A352992 a(4) = 753 because 753^2 = 426957777;
%e A352992 a(5) = 60753 because 60753^3 = 224234888577777.
%e A352992 Table with a(n) and A225401(n-1)
%e A352992    ---------------------------------------------------------------------------
%e A352992    |    |     a(n)       |      a'(n)        | A225401(n-1) |  concatenation |
%e A352992    | n  | with "exactly" | without "exactly" |  = b(n-1)    |  b(n-1)...b(0) |
%e A352992    ---------------------------------------------------------------------------
%e A352992      0           1                 1
%e A352992      1           3                 3               3                  ...3
%e A352992      2          53                53               5                 ...53
%e A352992      3        1753               753               7                ...753
%e A352992      4         753               753               0               ...0753
%e A352992      5       60753             60753               6              ...60753
%e A352992      6      660753            660753               6             ...660753
%e A352992      7     9660753           9660753               9            ...9660753
%e A352992   ..........................................................................
%e A352992 Also, as A225401(23) = 0, we have from a(21) up to a(25):
%e A352992 a(21) =     559631497233899660753;
%e A352992 a(22) =    3559631497233899660753;
%e A352992 a(23) =  193559631497233899660753, found by _Marius A. Burtea_;
%e A352992 a(24) =   93559631497233899660753;
%e A352992 a(25) = 2093559631497233899660753.
%p A352992 f:= proc(n) local t,x;
%p A352992        t:= 7/9*(10^n-1);
%p A352992        x:= rhs(op(msolve(x^3=t,10^n)));
%p A352992        while x^3 mod 10^(n+1) = 10*t+7 do x:= x + 10^n od;
%p A352992        x
%p A352992 end proc:
%p A352992 f(0):= 1:
%p A352992 map(f, [$0..30]); # _Robert Israel_, Jul 29 2025
%o A352992 (Python)
%o A352992 def a(n):
%o A352992     k, s, target = 1, "1", "7"*n
%o A352992     while s.rstrip("7") + target != s: k += 1; s = str(k**3)
%o A352992     return k
%o A352992 print([a(n) for n in range(8)]) # _Michael S. Branicky_, Apr 14 2022
%Y A352992 Cf. A000578, A017307, A039685, A225401, A353003.
%K A352992 nonn,base
%O A352992 0,2
%A A352992 _Bernard Schott_, Apr 14 2022
%E A352992 a(8)-a(9) from _Marius A. Burtea_, Apr 14 2022