A352992 -id:A352992 - cp's OEIS frontend

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

Bernard Schott, Apr 24 2022

nonn

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.

			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
  ------------------------------------------------------------------------------

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

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

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

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

A353003 Indices k where A225401(k) = 0.

Original entry on oeis.org

3, 23, 26, 32, 42, 46, 48, 51, 54, 84, 100, 101, 119, 123, 136, 184, 185, 202, 206, 216, 241, 263, 265, 272, 273, 284, 293, 323, 325, 332, 351, 352, 362, 392, 400, 406, 408, 410, 425, 432, 447, 449, 450, 466, 484, 488, 490, 497, 498, 509, 512, 522, 532, 534, 573, 585, 593, 595

Offset: 1

Michel Marcus, Apr 15 2022

These are also the indices k such that A352992(k+1) < A352992(k).

			3 is a term because A225401(3) = 0; and we have A352992(4)=753 < A352992(3)=1753.

Cf. A225401, A352992.

lista(nn) = {my(n=0, list = List()); for(i=1, nn, m=7*(10^i-1)/9; for(x=0, 9, if(((n+(x*10^(i-1)))^3)%(10^i)==m, n=n+(x*10^(i-1)); if (x==0, listput(list, i-1)); break;););); Vec(list);} \\ after A225401

cp's OEIS Frontend

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

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