A353384 Irregular triangle T(n,k) with row n listing A003592(j) not divisible by 20 such that A352218(A003592(j)) = n.

1, 2, 4, 5, 10, 8, 16, 25, 50, 32, 64, 125, 250, 128, 256, 625, 1250, 512, 1024, 3125, 6250, 2048, 4096, 15625, 31250, 8192, 16384, 78125, 156250, 32768, 65536, 390625, 781250, 131072, 262144, 1953125, 3906250, 524288, 1048576, 9765625, 19531250, 2097152, 4194304, 48828125, 97656250

Offset: 0

Author

Michael De Vlieger, Apr 15 2022

Keywords

• nonn
• easy
• base
• tabf

Comments

All terms in A003592 are products T(n,k)*20^j, j >= 0.

When expressed in base 20, T(n,k) does not end in zero, yet 1/T(n,k) is a terminating fraction, regular to 20.

The first 5 terms are the proper divisors of 20.

For these reasons, the terms may be called vigesimal "proper regular" numbers.

Examples

			Row 0 contains 1 since 1 is the empty product.
Row 1 contains 2, 4, 5, and 10 since these divide 20 and are not divisible by 20.
Row 2 contains 8, 16, 25, and 50 since these divide 20^2 but not 20. The other divisors of 20^2 either divide smaller powers of 20 or they are divisible by 20 and do not appear.
Row 3 contains 32, 64, 125, and 250 since these divide 20^3 but not 20^2. The other divisors of 20^3 either divide smaller powers of 20 or they are divisible by 20 therefore do not appear.
		

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..5722
• Eric Weisstein's World of Mathematics, Vigesimal
• Wikipedia, Vigesimal

Crossrefs

Cf. A003592, A352218.

Programs

Mathematica

{{1}}~Join~Array[Union@ Flatten@ {#, 2 #} &@ {2^(2 # - 1), 5^#} &, 11] // Flatten

Formula

Row 0 contains the empty product, thus row length = 1.

Row n sorts {2^(2n-1), 5^n, 2^(2n), 2*5^n}, thus row length = 4.