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

1,3

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, ...).

Links

Table of n, a(n) for n=1..47.

Formula

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

Example

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}.

Prog

(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

Crossrefs

Cf. A004290, A079339, A257345.

Subsequences: A131577, A175805.

Sequence in context: A174838 A308149 A196871 * A001856 A328265 A002081

Adjacent sequences: A334420 A334421 A334422 * A334424 A334425 A334426

Keyword

nonn,base

Author

Bernard Schott, May 25 2020

Status

approved