bibkey: frohlich2017a254074
authors: Felix Fröhlich; Chai Wah Wu
year: 2017
title: "OEIS A254074, Numbers k such that the decimal expansions of both k and k^2 have 5 as the digit with the smallest value and 9 as the digit with the largest value"
doi: null
url: https://oeis.org/A254074
claim: "A254074 %N: Numbers k such that the decimal expansions of both k and k^2 have 5 as the digit with the smallest value and 9 as the digit with the largest value. A254074 %C: The first digit of a term is either 7, 8 or 9 and the last digit is either 5, 6 or 7. Conjecture: no term is a multiple of 5. - _Chai Wah Wu_, Sep 10 2017"
strata_touched:
  - D5/S1/Digit/Admissibility/WuDigitRangeSquareNotMultipleOfFive
license: citation-only
triage: anchor

OEIS A254074

The NAME of A254074 states:

Numbers k such that the decimal expansions of both k and k^2 have 5 as the digit with the smallest value and 9 as the digit with the largest value.

Chai Wah Wu’s COMMENT of September 10, 2017 states:

The first digit of a term is either 7, 8 or 9 and the last digit is either 5, 6 or 7. Conjecture: no term is a multiple of 5. - Chai Wah Wu, Sep 10 2017

The AUTHOR line names:

Felix Fröhlich, May 03 2015

The sequence author is listed first in the note metadata, followed by the conjecture author. The year 2017 records the conjecture; the sequence AUTHOR line is dated May 3, 2015.

Verified locator

• URL: https://oeis.org/A254074
• NAME (%N, verbatim): Numbers k such that the decimal expansions of both k and k^2 have 5 as the digit with the smallest value and 9 as the digit with the largest value.
• COMMENT (%C, Sep 10 2017, verbatim): The first digit of a term is either 7, 8 or 9 and the last digit is either 5, 6 or 7. Conjecture: no term is a multiple of 5. - Chai Wah Wu, Sep 10 2017
• AUTHOR (%A, verbatim): Felix Fröhlich, May 03 2015

Only the second sentence of Wu’s comment, “Conjecture: no term is a multiple of 5,” is settled. The preceding first-digit and last-digit assertion is not claimed.

The route assumes a multiple of 5 whose digits are all at least 5, forcing its last digit to be 5. Writing the number as 10q+5 gives a square congruent to 25 modulo 100; once the number is at least 59, the tens digit 2 is present in the square and contradicts minimum digit 5.

An exhaustive scan through 12,000,000 found exactly one sequence term, 759576, whose square is 576955699776, and found no term divisible by 5. The single observed term makes this bounded scan thin, and the scan carries no proof of the universal statement.