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A294497 Squares k (not ending in 0) such that the integer that is built up by concatenating the floors of the square roots of the two-digit numbers into which the original number is separated (from right to left) is the square root of the original number. 1
1, 4, 9, 16, 25, 36, 49, 64, 81, 225, 625, 1225, 2025, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 4225, 5625, 7225, 9025, 22801, 23104, 23409, 50625, 63001, 63504, 75625, 123201, 180625, 203401, 225625, 390625, 432964, 455625, 573049, 680625, 732736, 765625, 2175625, 6260004, 6270016 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

If k has an odd number of digits, all digits after the first digit are paired; see first example below.

LINKS

David A. Corneth, Table of n, a(n) for n = 1..114

EXAMPLE

75625 is a term because partitioning its digits as (7)(56)(25), taking the square root of each part and truncating, and then concatenating the results, gives floor(sqrt(7))|floor(sqrt(56))|floor(sqrt(25)) = 275 = sqrt(75625);

180625 is a term because floor(sqrt(18))|floor(sqrt(06))|floor(sqrt(25)) = 425 = sqrt(180625).

MATHEMATICA

#^2 & /@ Select[Range[10^4], And[Mod[#, 10] != 0, FromDigits@ Map[Floor@ Sqrt@ FromDigits@ # &, Partition[PadLeft[#, 2 Ceiling[Length@ #/2]], 2, 2]] &@ IntegerDigits[#^2] == #] &] (* Michael De Vlieger, Nov 23 2017 *)

PROG

(PARI) is(n) = if(issquare(n) == 0||n % 10 == 0, return(0)); my(sq = i = 0, cn = n); while(cn > 0, sq += 10^i * sqrtint(cn % 100); cn \= 100; i++); sq ^ 2 == n \\ David A. Corneth, Jan 18 2018

(Python)

import math

for k in range(1, 1000000000):

   p = 0

   z = 0

   n = k*k

   while n >= 100:

      z = z + int(math.floor(math.sqrt(n % 100)) * math.pow(10, p))

      n = int((n - (n % 100)) / 100)

      p = p + 1

   z = z + int(math.floor(math.sqrt(n)) * math.pow(10, p))

   if z == k and k % 10 > 0:

      print(k * k, k)

CROSSREFS

Sequence in context: A078255 A077356 A077357 * A080160 A110723 A084617

Adjacent sequences:  A294494 A294495 A294496 * A294498 A294499 A294500

KEYWORD

nonn,base

AUTHOR

Reiner Moewald, Nov 01 2017

STATUS

approved

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Last modified August 18 20:02 EDT 2019. Contains 326109 sequences. (Running on oeis4.)