A049343 Numbers m such that 2m and m^2 have same digit sum.

0, 2, 9, 11, 18, 20, 29, 38, 45, 47, 90, 99, 101, 110, 119, 144, 146, 180, 182, 189, 198, 200, 245, 290, 299, 335, 344, 351, 362, 369, 380, 398, 450, 452, 459, 461, 468, 470, 479, 488, 495, 497, 639, 729, 794, 839, 848, 900, 929, 954, 990, 999

Offset: 1

Author

R. K. Guy

Keywords

• nonn
• base
• easy
• nice
• look

Comments

An easy way to prove that this sequence is infinite is to observe that it contains all numbers of the form 10^k+1. - Stefan Steinerberger, Mar 31 2006

For n>0: digital root (A010888) of 2n or n^2 is either 4 or 9. - Reinhard Zumkeller, Oct 01 2007

References

• Problem 117 in Loren Larson's translation of Paul Vaderlind's book.

Links

• Reinhard Zumkeller and Harvey P. Dale, Table of n, a(n) for n = 1..1000 (first 101 terms from Zumkeller)

Crossrefs

Cf. A058369, A077436 (binary). - Reinhard Zumkeller, Apr 03 2011

Programs

Haskell

import Data.List (elemIndices)
import Data.Function (on)
a049343 n = a049343_list !! (n-1)
a049343_list = elemIndices 0
   $ zipWith ((-) `on` a007953) a005843_list a000290_list
-- Reinhard Zumkeller, Apr 03 2011

Magma

[n: n in [0..1000] | &+Intseq(2*n) eq &+Intseq(n^2)]; // Vincenzo Librandi, Nov 17 2015

Mathematica

Select[Range[0, 1000], Sum[DigitCount[2# ][[i]]*i, {i, 1, 9}] == Sum[DigitCount[ #^2][[i]]*i, {i, 1, 9}] &] (* Stefan Steinerberger, Mar 31 2006 *)
Select[Range[0,1000],Total[IntegerDigits[2#]]==Total[ IntegerDigits[ #^2]]&] (* Harvey P. Dale, Sep 25 2012 *)

Formula

A007953(A005843(a(n))) = A007953(A000290(a(n))). - Reinhard Zumkeller, Oct 01 2007