A334423 Fixed points of A257345.

0, 1, 2, 4, 8, 16, 21, 32, 42, 64, 84, 128, 168, 256, 336, 512, 672, 1024, 1344, 2048, 2231, 2688, 4096, 4462, 5376, 8192, 9324, 10752, 16384, 18648, 21504, 32768, 37296, 43008, 65536, 74592, 86016, 131072, 149184, 172032, 262144, 298368, 344064, 524288, 596736, 688128, 1048576

Offset: 1

Author

Bernard Schott, May 25 2020

Keywords

• nonn
• base

Comments

The least positive multiple of an integer m that when written in base 10 uses only 0's and 1's is q = A004290(m) = k*m. If we regard q as binary number and converts q to base 10, we get A257345(q) = u. When m = u, then m is a term.

If m is a term, then m*2^k is another term.

The first 3 primitive terms are 1, 21, 2231 and the 3 corresponding subsequences of such fixed points are,

-> m = 0 or m = 2^k, k>=0 (A131577),

-> m = 21 * 2^k, k>=0 (A175805),

-> m = 2231 * 2^k, k>=0 (2231, 4462, 9324, 18648, ...).

Examples

			The least positive multiple of 42 that when written in base 10 uses only 0's and 1's is 101010 = 2405*42. If we regard 101010 as binary number and converts to base 10, we get 42; hence, 42 is a term.
Successive operations for first primitive terms:
1 --> A004290(1) = 1_{10} --> 1_{2} = 1_{10},
21 --> A004290(21) = 10101_{10} --> 10101_{2} = 21_{10},
2231 --> A004290(2231) = 100010110111_{10} --> 100010110111_{2} = 2231_{10}.
		

Crossrefs

Cf. A004290, A079339, A257345.

Subsequences: A131577, A175805.

Programs

PARI

f(n) = {if( n==0, return (0)); my(m = n); while (vecmax(digits(m)) != 1, m+=n); m; } \\ A004290
isok(m) = fromdigits(digits(f(m), 10), 2) == m; \\ Michel Marcus, May 29 2020

Formula

A257345(A004290(a(n))) = a(n).