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%I #32 May 06 2022 13:13:51
%S 1,7,77,477,6477,46477,446477,5446477,85446477,385446477,4385446477,
%T 44385446477,644385446477,8644385446477,38644385446477,
%U 138644385446477,5138644385446477,115138644385446477,15138644385446477,5015138644385446477
%N Smallest positive integer whose cube ends with exactly n 3's.
%C 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 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 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 There are similar sequences when cubes end with 1, 7, 8 or 9.
%F When n is not in A352282, a(n) = Sum_{k=0..n-1} A225402(k) * 10^k.
%e a(0) = 1 because 1^3 = 1;
%e a(1) = 7 because 7^3 = 343;
%e a(2) = 77 because 77^3 = 456533;
%e a(3) = 477 because 477^3 = 108531333;
%e ------------------------------------------------------------------------------
%e | | a(n) | a'(n) | A225402(n-1) | concatenation |
%e | n | with "exactly" | without "exactly" | = b(n-1) | b(n-1)...b(0) |
%e ------------------------------------------------------------------------------
%e 1 7 7 7 ...7
%e 2 77 77 7 ...77
%e 3 477 477 4 ...477
%e ............................................................................
%e 15 138644385446477 138644385446477 1 ...138644385446477
%e 16 5138644385446477 5138644385446477 5 ...5138644385446477
%e 17 115138644385446477 15138644385446477 1 ...15138644385446477
%e 18 15138644385446477 15138644385446477 0 ...015138644385446477
%e 19 5015138644385446477 5015138644385446477 5 ...5015138644385446477
%e ------------------------------------------------------------------------------
%Y Cf. A225402, A352282, A352992 (similar, with 7).
%K nonn
%O 0,2
%A _Bernard Schott_, Apr 24 2022