cp's OEIS Frontend

A259316 Numbers n such that the result of n multiplied by the reversal of n can be split into two numbers a and b of equal length (if the length is odd a leading zero is allowed), where a + b equals n (b can also have a leading zero).

%I A259316 #39 Sep 26 2023 11:55:50
%S A259316 1,9,54,55,99,999,2727,3222,7777,8272,9999,12466,22222,25912,39114,
%T A259316 75880,87777,87804,93357,99999,124660,142857,181818,185185,189189,
%U A259316 230769,231868,324675,390313,412587,428274,443926,503866,513513,533169,568468
%N A259316 Numbers n such that the result of n multiplied by the reversal of n can be split into two numbers a and b of equal length (if the length is odd a leading zero is allowed), where a + b equals n (b can also have a leading zero).
%C A259316 All rep'n'-digits have infinite subsequence, except the rep'n'-digits 3 (mod 9) and 6 (mod 9).
%C A259316 For 'n' is 1, we have the Kaprekar numbers (A145875), the repdigit numbers.
%C A259316 If length is 1 (mod 9), repdigit 1 is part of the sequence, 1111111111*1111111111 = 1234567900987654321 => 123456790 + 987654321 = 1111111111.
%C A259316 If length is 2 (mod 9), repdigit 5 is part of the sequence, 55555555555*55555555555 = 3086419753024691358025 => 30864197530 + 24691358025 = 5555555555.
%C A259316 If length is 4 (mod 9), repdigit 7 is part of the sequence, 7777 * 7777 = 60481729 => 6048 + 1729 = 7777.
%C A259316 If length is 5 (mod 9), repdigit 2 is part of the sequence.
%C A259316 If length is 7 (mod 9), repdigit 4 is part of the sequence.
%C A259316 If length is 8 (mod 9), repdigit 8 is part of the sequence.
%C A259316 Repdigit 9 is part of this sequence in every length.
%C A259316 For 'n' is 2, we have numbers where two digits are repeated, like 52525252.
%C A259316 The rep2-digits which are divisible by 9 have the following infinite subsequences:
%C A259316 If length is 2 (mod 22), rep2-digit 54 is a part of this sequence, 545454545454545454545454 * 454545454545454545454545 = 247933884297520661157024297520661157024793388430 => 247933884297520661157024 + 297520661157024793388430 = 545454545454545454545454
%C A259316 If length is 4 (mod 22), rep2-digit 27 is a part of this sequence.
%C A259316 If length is 6 (mod 22), rep2-digit 18 is a part of this sequence.
%C A259316 If length is 8 (mod 22), rep2-digit 63 is a part of this sequence.
%C A259316 If length is 10 (mod 22), rep2-digit 90 is a part of this sequence.
%C A259316 If length is 14 (mod 22), rep2-digit 36 is a part of this sequence.
%C A259316 If length is 16 (mod 22), rep2-digit 81 is a part of this sequence.
%C A259316 If length is 18 (mod 22), rep2-digit 72 is a part of this sequence.
%C A259316 If length is 20 (mod 22), rep2-digit 45 is a part of this sequence.
%C A259316 Other rep2-digits also have infinite subsequences with length l (mod 198).
%C A259316 Example: rep2-digit 52 has length 8: 52525252 * 25252525 = 1326395239261300 => 13263952 + 39261300 = 52525252, the next length is 206.
%H A259316 Pieter Post and Giovanni Resta, <a href="/A259316/b259316.txt">Table of n, a(n) for n = 1..187</a> (terms < 10^12, first 84 terms from Pieter Post)
%e A259316 124660 is a term. Indeed 124660*66421 = 8280041860 and 82800 + 41860 = 124660.
%t A259316 fQ[n_] := Block[{c, d, len}, c = n FromDigits@ Reverse@ IntegerDigits@ n; d = IntegerDigits@ c; len = Length@ d; If[OddQ@ len, d = PadLeft[d, len + 1]; len++]; n == FromDigits@ Take[d, len/2] + FromDigits@ Take[d, -len/2]]; Select[Range@ 1000000, fQ] (* _Michael De Vlieger_, Jul 20 2015 *)
%o A259316 (Python)
%o A259316 def sod(n,m):
%o A259316     kk = 0
%o A259316     while n > 0:
%o A259316         kk= kk+(n%m)
%o A259316         n =int(n//m)
%o A259316     return kk
%o A259316 uu=1
%o A259316 for a in range (1,9):
%o A259316     for n in range (10**(a-1)+1, 10**a):
%o A259316         y=int(str(n)[::-1])
%o A259316         ll=int(len(str(n*y))/2+0.5)
%o A259316         u=sod(n*y,10**ll)
%o A259316         if n==u:
%o A259316             print (n)
%o A259316 (Python)
%o A259316 # for rep2-digit
%o A259316 for f in range (12,98):
%o A259316     aa=1
%o A259316     for i in range(1,200):
%o A259316         aa=10**(2*i)+aa
%o A259316         c=f*aa
%o A259316         cc=str(c*int(str(c)[::-1]))
%o A259316         l=int(len(cc)/2)
%o A259316         cc1,cc2=int(cc[0:l]),int(cc[l:2*l+1])
%o A259316         if c==cc1+cc2:
%o A259316             print (c)
%Y A259316 Cf. A145875, A064942, A053816, A006886.
%K A259316 nonn,base
%O A259316 1,2
%A A259316 _Pieter Post_, Jun 24 2015
%E A259316 Missing a(21) from _Giovanni Resta_, Jul 19 2015