A378138 -id:A378138 - cp's OEIS frontend

A381087 The smallest positive integer that produces a product that contains the digit 2 when multiplied by 2 a total of n times.

2, 1, 6, 31, 64, 64, 331, 331, 814, 1607, 4107, 5129, 5129, 5129, 10283, 12819, 16163, 16163, 16163, 40108, 40108, 40108, 40108, 40108, 40108, 80313, 80313, 80313, 80313, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153

Offset: 0

Scott R. Shannon, Feb 13 2025

The last known distinct term is a(148) = 3130008; all subsequent terms studied also equal 3130008, and it is plausible, although unproven, that this is the last distinct value as n -> infinity.

			a(2) = 6 as 6*2 = 12 and 12*2 = 24, and the two products contain the digit 2.
a(8) = 814 as 814*2 = 1628, 1628*2 = 3256, 3256*2 = 6512, 6512*2 = 13024, 13024*2 = 26048, 26048*2 = 52096, 52096*2 = 104192, 104192*2 = 208384, and the eight products contain the digit 2.

Scott R. Shannon, Table of n, a(n) for n = 0..500

Cf. A378138 (distinct values), A381183, A011532.

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.

Original entry on oeis.org

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

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

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.

			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.

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

Cf. A381087, A378138, A011532.

cp's OEIS Frontend

A381087 The smallest positive integer that produces a product that contains the digit 2 when multiplied by 2 a total of n times.

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