A299539 Numbers n = d_1 d_2 ... d_k (in base 10) such that d_i + d_{k+1-i} = 10 for i = 1..k.

5, 19, 28, 37, 46, 55, 64, 73, 82, 91, 159, 258, 357, 456, 555, 654, 753, 852, 951, 1199, 1289, 1379, 1469, 1559, 1649, 1739, 1829, 1919, 2198, 2288, 2378, 2468, 2558, 2648, 2738, 2828, 2918, 3197, 3287, 3377, 3467, 3557, 3647, 3737, 3827, 3917, 4196, 4286

Offset

1,1

Comments

These numbers are also called upside-down numbers.

All terms belong to A052382 (zeroless numbers).

The central digit of the terms with an odd number of digits is always 5.

This sequence can be partitioned into three sets: { 5 }, A083678 and A093472.

This sequence has similarities with A284811: here d_i + d_{k+1-i} = 10, there d_i + d_{k+1-i} = 9.

These numbers have a palindromic Morse code representation (see A060109). To get all numbers with this property one has to include 0 and terms with corresponding "interior" digits 5 replaced by digits 0, e.g., 5 -> 0, 159 -> 109, 555 -> 505, 1559 -> 1009, 15559 -> {10009, 10509, 15059}. - M. F. Hasler, Nov 02 2020

Links

Table of n, a(n) for n=1..48.

Robert E. Kennedy and Curtis N. Cooper, Bach, 5465, and Upside-Down Numbers, The College Mathematics Journal, Vol. 18, No. 2 (Mar., 1987), pp. 111-115.

Giovanni Resta, Upside-down numbers, Numbers Aplenty.

Example

1 + 9 = 10 and 5 + 5 = 10 and 9 + 1 = 10, hence 159 belongs to this sequence.
4 + 2 = 6, hence 42 does not belong to this sequence.

Maple

Res:= NULL;
for d from 1 to 3 do
  for x from 0 to 9^(d-1)-1 do
    L:= convert(9^(d-1)+x, base, 9)[1..d-1];
    Res:= Res, 5*10^(d-1)+add((1+L[-i])*10^(2*d-1-i)+(9-L[-i])*10^(i-1), i=1..d-1)
  od;
  for x from 0 to 9^d-1 do
    L:= convert(9^d+x, base, 9)[1..d];
    Res:= Res, add((1+L[-i])*10^(2*d-i)+(9-L[-i])*10^(i-1), i=1..d)
  od
od:
Res; # Robert Israel, Mar 06 2018

Mathematica

Select[Range[4300], AllTrue[#1[[1 ;; #2]] + Reverse@ #1[[-#2 ;; -1]], # == 10 &] & @@ {#, Ceiling[Length[#]/2]} &@ IntegerDigits[#] &] (* Michael De Vlieger, Nov 04 2020 *)

Prog

(PARI) is(n) = my (d=digits(n)); Set(d+Vecrev(d))==Set(10)

Crossrefs

Cf. A052382, A083678, A093472, A284811.

Cf. also A060109 (Morse code of numbers).

Sequence in context: A332155 A366211 A061388 * A270865 A106072 A106062

Adjacent sequences: A299536 A299537 A299538 * A299540 A299541 A299542

Keyword

nonn,base,easy

Author

Rémy Sigrist, Mar 05 2018

Status

approved