A391471 Integers k such that floor((11/10)*k + 1/2) is a digit permutation of k.

1, 2, 3, 4, 89, 179, 269, 359, 449, 539, 629, 719, 809, 895, 896, 897, 898, 899, 1348, 1795, 1796, 1797, 1798, 1799, 2247, 2695, 2696, 2697, 2698, 2699, 3146, 3595, 3596, 3597, 3598, 3599, 4045, 4495, 4496, 4497, 4498, 4499, 4590, 4954, 5395, 5396, 5397, 5398, 5399, 6295, 6296, 6297, 6298, 6299

Offset

1,2

Comments

Motivation: observed from 10% consumption-tax price rounding.

Leading zeros are not allowed.

If T = floor((11/10)*n + 1/2), then necessary conditions for a k-digit solution are: (i) T < 10^k (no carry to a new digit), (ii) T == n (mod 9), (iii) the multisets of digits of n and T coincide.

Links

Masato Wada, Table of n, a(n) for n = 1..484.

Formula

Let T(n) = floor((11*n + 5)/10). Then n is a term iff length(n) = length(T(n)) and sort(digits(n)) = sort(digits(T(n))).

Example

3598 is a term since T(3598) = floor((11/10)*3598 + 0.5) = 3958, a digit permutation of 3598.
89 is a term since T(89) = 98.

Maple

taxround := n -> floor(11*n/10 + 1/2):
isA := proc(n) local t, dn, dt;
  t  := taxround(n);
  dn := convert(n, base, 10); dt := convert(t, base, 10);
  if nops(dn) <> nops(dt) then return false fi;
  sort(dn) = sort(dt)
end:
select(isA, [$1..10000]);

Mathematica

T[n_] := Floor[11 n/10 + 1/2];
isA[n_] := With[{t = T[n]},
  IntegerLength[n] == IntegerLength[t] &&
  Sort[IntegerDigits[n]] === Sort[IntegerDigits[t]]
];
Select[Range[1, 10000], isA]

Prog

(Python)
def T(n): return (11*n + 5)//10
def ok(n):
    s, t = str(n), str(T(n))
    return len(s) == len(t) and sorted(s) == sorted(t)
print([k for k in range(1, 10**4) if ok(k)])
(Python)
from itertools import count, islice
def A391471_gen(): # generator of terms
    return filter(lambda n:sorted(str(n))==sorted(str(11*n+5)[:-1]), count(1))
A391471_list = list(islice(A391471_gen(), 54)) # Chai Wah Wu, Dec 19 2025
(PARI)
T(n)=(11*n+5)\10;
isA(n)=
{
  my(t=T(n));
  if(#digits(n)!=#digits(t), return(0));
  vecsort(digits(n))==vecsort(digits(t))
};
select(isA, [1..10000])

Crossrefs

Sequence in context: A037395 A009496 A263281 * A191422 A008405 A037431

Adjacent sequences: A391468 A391469 A391470 * A391472 A391473 A391474

Keyword

nonn,base

Author

Masato Wada, Dec 14 2025

Status

approved