A146754 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 5.

142857, 142857142857, 142857142857142857, 142857142857142857142857, 142857142857142857142857142857, 142857142857142857142857142857142857, 102040816326530612244897959183673469387755, 122448979591836734693877551020408163265306, 142857142857142857142857142857142857142857

Offset: 1

Author

N. J. A. Sloane, based on correspondence from William A. Hoffman III (whoff(AT)robill.com), Apr 10 2009

Keywords

• nonn
• base

Comments

From Seiichi Manyama, Aug 22 2017: (Start)

For k >= 1, (10^(6*k) - 1)/7 is a term.

For 5 <= a <= 9 and k >= 1, a*(10^(42*k) - 1)/49 is a term. (End)

For consistency with A146088 (similar for ratio 2), where an initial a(0) = 0 has been added, the same could be considered here. It would be compatible with the formulas given above (with k = 0). - M. F. Hasler, May 03 2025

Examples

			From _Seiichi Manyama_, Aug 22 2017: (Start)
a(1) = b1*10 + 7 with b1 = 14285, and 5*a(1) = 714285 = 7*10^5 + b1.
a(7) = b7*10 + 5 with b7 = 10204081632653061224489795918367346938775, and
  5*a(7) = 510204081632653061224489795918367346938775 = 5*10^41 + b7. (End)
		

Links

• Robert Israel, Table of n, a(n) for n = 1..258
• Wikipedia, Parasitic number.

Crossrefs

Cf. A146088 (k=2), A146561 (k=3), A146569 (k=4), this sequence (k=5), A291354 (k=6), A291215 (k=7), A291321 (k=8), A291353 (k=9).

All these are subsequences of A034089 (except for an initial 0 in some of them).

Cf. A092697, A097717.

Programs

Maple

f:= proc(d) # solutions with d+1 digits
    local b,R,a;
    R:= NULL;
    for b from ceil(49*10^(d-1)/(10^d - 1)) to 9 do
       a:= (10^d-5)*b/49;
       if a::integer then R:= R, 10*a+b fi
    od;
   R
end proc:
map(f, [$1..42]); # Robert Israel, Nov 05 2024