A381183 a(n) = the smallest positive integer that produces a product that contains the digit 2 when multiplied by 2 at most n times, and where a further multiplication by 2 produces a number that does not contain the digit 2. Set a(n) = -1 if no such number exists.

2, 1, 6, 31, 128, 64, 516, 331, 814, 1607, 4107, 10158, 10258, 5129, 10283, 12819, 25633, 28141, 16163, 51404, 80134, 80864, 40633, 80216, 40108, 128129, 250627, 160626, 80313, 125641, 208141, 383814, 391628, 195814, 156766, 196314, 391563, 490641, 806166, 785313, 628222, 314111, 625322, 312661, 1563305, 2630104, 1315052, 657526, 328763, 1643815

Offset: 0

Author

Michael De Vlieger and Scott R. Shannon, Feb 16 2025

Keywords

• nonn
• base

Comments

It is plausible that there are many terms, likely almost all terms, such that a(n) = -1, since the products as n increases become so large it is almost certain that subsequent products also contain the digit 2. It is therefore extremely unlikely that the series of products will terminate for very large values of n. See A381087.

For all starting values up to 10^9 the lowest undetermined term is a(263), while the largest determined term is a(370) = 357131067. The largest term value in this range is a(301) = 957107659.

Examples

			a(2) = 6 as 6*2 = 12, 12*2 = 24, 24*2 = 48, and the first two products contain the digit 2 while the third does not.
a(6) = 516 as 516*2 = 1032, 1032*2 = 2064, 2064*2 = 4128, 4128*2 = 8256, 8256*2 = 16512, 16512*2 = 33024, 33024*2 = 66048, and the first six products contain the digit 2 while the seventh does not.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 0..262

Crossrefs

Cf. A381087, A378138, A011532.