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A334423
Fixed points of A257345.
0
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, ...).
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
Subsequences: A131577, A175805.
Sequence in context: A174838 A308149 A196871 * A375006 A001856 A328265
KEYWORD
nonn,base
AUTHOR
Bernard Schott, May 25 2020
STATUS
approved