A384131 Smallest positive number divisible by n that has n letters in US English, or 0 if none exists.

6, 4, 40, 12, 70, 56, 36, 100, 33, 300, 1000000001, 406, 150, 112, 170, 162, 418, 11020, 336, 528, 828, 4800, 3300, 1404, 1620, 1512, 1218, 1770, 1147, 1344, 1353, 2788, 3325, 3888, 12728, 13376, 13338, 103360, 22878, 23478, 27778, 101728, 103725, 111734, 111578

Offset: 3

Author

Jason Bard, May 20 2025

Keywords

• nonn
• word

Comments

The GCHQ reference uses British English (including "and"). The disagreements are at a(14)-a(16), which the book lists as 70000000, 15000, 14000. Furthermore, the book lists a(13) as impossible.

References

• GCHQ, The GCHQ Puzzle Book, Penguin, 2016 (see p. 47).

Links

• Michael S. Branicky, Table of n, a(n) for n = 3..100 (terms 3..69 from Jason Bard)

Crossrefs

Cf. A005589, A134629.

Programs

Mathematica

mmax = 10^10; Do[m = n; While[StringLength[StringDelete[IntegerName[m, "Words"], {" ", "-", "\[Hyphen]", ","}]] != n, m += n; If[m > mmax, m = 0; Break[]]]; Print[m], {n, 3, 43}]

Python

from num2words import num2words
from itertools import count, islice
def b(n): return sum(1 for c in num2words(n).replace(" and", "") if c.isalpha())
def a(n): return next(k for k in count(n, n) if b(k) == n)
print([a(n) for n in range(3, 13)]) # Michael S. Branicky, May 20 2025

Formula

a(n) >= A134629(n). - Michael S. Branicky, May 21 2025