A166749 Numbers that are the sum or product of two numbers, such that the sum and product have reversed digits.

0, 4, 18, 27, 49, 72, 81, 94, 499, 994, 4999, 9994, 49999, 99994, 499999, 999994, 4999999, 9999994, 49999999, 99999994, 499999999, 999999994, 4999999999, 9999999994, 49999999999, 99999999994, 499999999999, 999999999994, 4999999999999, 9999999999994, 49999999999999

Offset: 1

Author

Mark Nandor, Oct 21 2009

Keywords

• nonn
• base
• easy

Comments

Note that 0 and 4 are their own reversed-digit sums and products, since 0+0=0*0=0 and 2+2=2*2=4. The pattern of some number of nines and then a four, and a four and some number of nines, continues indefinitely.

These are in fact all the solutions, shown by a case-by-case analysis. - Wang Pok Lo, Dec 24 2018

Examples

			For instance, 9*9=81 and 9+9=18 are terms; 3*24=72 and 3+24=27 are terms too.
		

Links

• W. P. Lo and Y. Paz, On finding all positive integers a,b such that b±a and ab are palindromic, arXiv:1812.08807 [math.HO] (2018).

Programs

Mathematica

Do[If[IntegerDigits[x y] == Reverse[IntegerDigits[x + y]], Print[{x, y, x + y, x y}]], {x, 0, 20}, {y, x, 100000}] or a[1]=0;a[2]=4;a[3]=18;a[4]=27;a[5]=49;a[6]=72;a[7]=81;a[8]=94 a[n_] := a[n] = If[OddQ[n], 5*10^((n + 1)/2 - 3) - 1, 10^(n/2 - 2) - 6]

Formula

For n>8, a(n)=5*10^((n+1)/2 - 3) - 1 if n odd; a(n)=10^(n/2 - 2) - 6 if n even.