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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). 1
1, 9, 54, 55, 99, 999, 2727, 3222, 7777, 8272, 9999, 12466, 22222, 25912, 39114, 75880, 87777, 87804, 93357, 99999, 124660, 142857, 181818, 185185, 189189, 230769, 231868, 324675, 390313, 412587, 428274, 443926, 503866, 513513, 533169, 568468 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

All rep'n'-digits have infinite subsequence, except the rep'n'-digits 3 (mod 9) and 6 (mod 9).

For 'n' is 1, we have the Kaprekar numbers (A145875), the repdigit numbers.

If length is 1 (mod 9), repdigit 1 is part of the sequence,

1111111111*1111111111 = 1234567900987654321 => 123456790 + 987654321 = 1111111111.

If length is 2 (mod 9), repdigit 5 is part of the sequence,

55555555555*55555555555 = 3086419753024691358025 => 30864197530 + 24691358025 = 5555555555.

If length is 4 (mod 9), repdigit 7 is part of the sequence, 7777 * 7777 = 60481729 => 6048 + 1729 = 7777.

If length is 5 (mod 9), repdigit 2 is part of the sequence.

If length is 7 (mod 9), repdigit 4 is part of the sequence.

If length is 8 (mod 9), repdigit 8 is part of the sequence.

Repdigit 9 is part of this sequence in every length.

For 'n' is 2, we have numbers where two digits are repeated, like 52525252.

The rep2-digits which are divisible by 9 have the following infinite subsequences:

If length is 2 (mod 22), rep2-digit 54 is a part of this sequence, 545454545454545454545454 * 454545454545454545454545 = 247933884297520661157024297520661157024793388430 => 247933884297520661157024 + 297520661157024793388430 = 545454545454545454545454

If length is 4 (mod 22), rep2-digit 27 is a part of this sequence.

If length is 6 (mod 22), rep2-digit 18 is a part of this sequence.

If length is 8 (mod 22), rep2-digit 63 is a part of this sequence.

If length is 10 (mod 22), rep2-digit 90 is a part of this sequence.

If length is 14 (mod 22), rep2-digit 36 is a part of this sequence.

If length is 16 (mod 22), rep2-digit 81 is a part of this sequence.

If length is 18 (mod 22), rep2-digit 72 is a part of this sequence.

If length is 20 (mod 22), rep2-digit 45 is a part of this sequence.

Other rep2-digits also have infinite subsequences with length l (mod 198).

Example: rep2-digit 52 has length 8: 52525252 * 25252525 = 1326395239261300 => 13263952 + 39261300 = 52525252, the next length is 206.

LINKS

Pieter Post and Giovanni Resta, Table of n, a(n) for n = 1..187 (terms < 10^12, first 84 terms from Pieter Post)

EXAMPLE

124660 is a term. Indeed 124660*66421 = 8280041860 and 82800 + 41860 = 124660.

MATHEMATICA

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 *)

PROG

(Python)

def sod(n, m):

....kk = 0

....while n > 0:

........kk= kk+(n%m)

........n =int(n//m)

....return kk

uu=1

for a in range (1, 9):

....for n in range (10**(a-1)+1, 10**a):

........y=int(str(n)[::-1])

........ll=int(len(str(n*y))/2+0.5)

........u=sod(n*y, 10**ll)

........if n==u:

............print (n)

(python for rep2-digit)

for f in range (12, 98):

....aa=1

....for i in range(1, 200):

........aa=10**(2*i)+aa

........c=f*aa

........cc=str(c*int(str(c)[::-1]))

........l=int(len(cc)/2)

........cc1, cc2=int(cc[0:l]), int(cc[l:2*l+1])

........if c==cc1+cc2:

............print (c)

CROSSREFS

Cf. A145875, A064942, A053816, A006886.

Sequence in context: A326606 A052108 A209453 * A224484 A225791 A093846

Adjacent sequences:  A259313 A259314 A259315 * A259317 A259318 A259319

KEYWORD

nonn,base

AUTHOR

Pieter Post, Jun 24 2015

EXTENSIONS

Missing a(21) from Giovanni Resta, Jul 19 2015

STATUS

approved

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Last modified February 19 18:44 EST 2020. Contains 332047 sequences. (Running on oeis4.)