A246532 Smallest Meertens number in base n, or -1 if none exists.

2, 10, 200, 6, 54, 100, 216, 4199040, 81312000

Offset: 2

Author

David Applegate, Aug 28 2014

Keywords

• sign
• more
• hard

Comments

A Meertens number in base n is a fixed point of the base n Godel encoding.

The base n Godel encoding of x is 2^d(1) * 3^d(2) * ... * prime(k)^d(k), where d(1)d(2)...d(k) is the base n representation of x.

a(14) = 47250, a(16) = 18.

In a computer search that included all numbers < 10^29 and bases <= 16, the only additional Meertens numbers found were 6 (base 2), 10 (base 2), 49000 (base 5), and 181400 (base 5).

There is no base 11 Meertens number < 11^44 ~= 6.6*10^45.

There is no base 12 Meertens number < 12^40 ~= 1.4*10^43.

There is no base 13 Meertens number < 13^39 ~= 2.7*10^43.

There is no base 15 Meertens number < 15^37 ~= 3.2*10^43.

Other terms: a(17) = 36, a(19) = 96, a(32) = 256, a(51) = 54. - Chai Wah Wu, Aug 28 2014

From Chai Wah Wu, Jul 20 2020: (Start)

All terms are even.

If n > 2 and a(n) != -1, then a(n) > n.

a(2*3^m-m) = 2*3^m for all m >= 0, i.e., a(n) > 0 for an infinite number of values of n.

Other terms: a(64) = a(4096) = 65536, a(71) = 216, a(160) = 324, a(323) = 1296, a(1455) = 2916, a(1942) = 5832, a(7775) = 46656, a(8294) = 82944, a(13118) = 26244.

(End)

Named by Bird (1998) after the Dutch computer scientist Lambert Meertens (b. 1944). - Amiram Eldar, Jun 23 2021

a(10) = 81312000 is the only base-10 Meertens number below 10^50. - Max Alekseyev, Jul 24 2024

Examples

			100 is a base 7 Meertens number because 100 = 202_7 = 2^2 * 3^0 * 5^2.
4199040 is a base 9 Meertens number because 4199040 = 7810000_9 = 2^7 * 3^8 * 5^1.
		

Links

• David Applegate, C++ program used to search for Meertens numbers
• Richard S. Bird, Functional Pearl: Meertens number, Journal of Functional Programming, Vol. 8, No. 1 (Jan 1998), pp. 83-88.
• Wikipedia, Meertens number.
• Chai Wah Wu, Meertens Number and Its Variations, IBM Research Report RC25531 (WAT1504-032), April 2015.
• Chai Wah Wu, Meertens Number and Its Variations, arXiv:1603.08493 [math.NT], 2016.

Crossrefs

Cf. A189398 (base 10 Godel encoding), A110765 (base 2 Godel encoding).

Extensions

Conjectural terms removed from the sequence by Max Alekseyev, Jul 22 2024