A352995 Smallest positive integer whose cube ends with exactly n 3's.

1, 7, 77, 477, 6477, 46477, 446477, 5446477, 85446477, 385446477, 4385446477, 44385446477, 644385446477, 8644385446477, 38644385446477, 138644385446477, 5138644385446477, 115138644385446477, 15138644385446477, 5015138644385446477

Offset: 0

Author

Bernard Schott, Apr 24 2022

Keywords

• nonn

Comments

When A225402(k) = 0, i.e., k is a term of A352282, then a(k) > a(k+1); 1st example is for k = 17 with a(17) = 115138644385446477 > a(18) = 15138644385446477; otherwise, a(n) < a(n+1).

When n <> k, a(n) coincides with the 'backward concatenation' of A225402(n-1) up to A225402(0), where A225402 is the 10-adic integer x such that x^3 = -1/3 (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 also: 1, 7, 77, 477, 6477, 46477, 446477, ..., first difference arrives for n = 17.

There are similar sequences when cubes end with 1, 7, 8 or 9.

Examples

			a(0) = 1 because 1^3 = 1;
a(1) = 7 because 7^3 = 343;
a(2) = 77 because 77^3 = 456533;
a(3) = 477 because 477^3 = 108531333;
  ------------------------------------------------------------------------------
  |    |     a(n)          |      a'(n)        | A225402(n-1) | concatenation  |
  | n  | with "exactly"    | without "exactly" |  = b(n-1)    | b(n-1)...b(0)  |
  ------------------------------------------------------------------------------
    1                    7                    7     7                      ...7
    2                   77                   77     7                     ...77
    3                  477                  477     4                    ...477
   ............................................................................
   15      138644385446477      138644385446477     1        ...138644385446477
   16     5138644385446477     5138644385446477     5       ...5138644385446477
   17   115138644385446477    15138644385446477     1      ...15138644385446477
   18    15138644385446477    15138644385446477     0     ...015138644385446477
   19  5015138644385446477  5015138644385446477     5    ...5015138644385446477
  ------------------------------------------------------------------------------
		

Links

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

Crossrefs

Cf. A225402, A352282, A352992 (similar, with 7).

Programs

Maple

f:= proc(n) local t,x;
       t:= 3/9*(10^n-1);
       x:= rhs(op(msolve(x^3=t,10^n)));
       while x^3 mod 10^(n+1) = 10*t+3 do x:= x + 10^n od;
       x
end proc:
f(0):= 1:
map(f, [$0..20]); # Robert Israel, Jul 29 2025

Formula

When n is not in A352282, a(n) = Sum_{k=0..n-1} A225402(k) * 10^k.