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

%I A352995 #36 Jul 29 2025 14:25:53
%S A352995 1,7,77,477,6477,46477,446477,5446477,85446477,385446477,4385446477,
%T A352995 44385446477,644385446477,8644385446477,38644385446477,
%U A352995 138644385446477,5138644385446477,115138644385446477,15138644385446477,5015138644385446477
%N A352995 Smallest positive integer whose cube ends with exactly n 3's.
%C A352995 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).
%C A352995 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.
%C A352995 Without "exactly" in the name, terms a'(n) should be also: 1, 7, 77, 477, 6477, 46477, 446477, ..., first difference arrives for n = 17.
%C A352995 There are similar sequences when cubes end with 1, 7, 8 or 9.
%H A352995 Robert Israel, <a href="/A352995/b352995.txt">Table of n, a(n) for n = 0..996</a>
%F A352995 When n is not in A352282, a(n) = Sum_{k=0..n-1} A225402(k) * 10^k.
%e A352995 a(0) = 1 because 1^3 = 1;
%e A352995 a(1) = 7 because 7^3 = 343;
%e A352995 a(2) = 77 because 77^3 = 456533;
%e A352995 a(3) = 477 because 477^3 = 108531333;
%e A352995   ------------------------------------------------------------------------------
%e A352995   |    |     a(n)          |      a'(n)        | A225402(n-1) | concatenation  |
%e A352995   | n  | with "exactly"    | without "exactly" |  = b(n-1)    | b(n-1)...b(0)  |
%e A352995   ------------------------------------------------------------------------------
%e A352995     1                    7                    7     7                      ...7
%e A352995     2                   77                   77     7                     ...77
%e A352995     3                  477                  477     4                    ...477
%e A352995    ............................................................................
%e A352995    15      138644385446477      138644385446477     1        ...138644385446477
%e A352995    16     5138644385446477     5138644385446477     5       ...5138644385446477
%e A352995    17   115138644385446477    15138644385446477     1      ...15138644385446477
%e A352995    18    15138644385446477    15138644385446477     0     ...015138644385446477
%e A352995    19  5015138644385446477  5015138644385446477     5    ...5015138644385446477
%e A352995   ------------------------------------------------------------------------------
%p A352995 f:= proc(n) local t,x;
%p A352995        t:= 3/9*(10^n-1);
%p A352995        x:= rhs(op(msolve(x^3=t,10^n)));
%p A352995        while x^3 mod 10^(n+1) = 10*t+3 do x:= x + 10^n od;
%p A352995        x
%p A352995 end proc:
%p A352995 f(0):= 1:
%p A352995 map(f, [$0..20]); # _Robert Israel_, Jul 29 2025
%Y A352995 Cf. A225402, A352282, A352992 (similar, with 7).
%K A352995 nonn
%O A352995 0,2
%A A352995 _Bernard Schott_, Apr 24 2022