cp's OEIS Frontend

A334423 Fixed points of A257345.

%I A334423 #31 Aug 31 2021 01:12:52
%S A334423 0,1,2,4,8,16,21,32,42,64,84,128,168,256,336,512,672,1024,1344,2048,
%T A334423 2231,2688,4096,4462,5376,8192,9324,10752,16384,18648,21504,32768,
%U A334423 37296,43008,65536,74592,86016,131072,149184,172032,262144,298368,344064,524288,596736,688128,1048576
%N A334423 Fixed points of A257345.
%C A334423 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.
%C A334423 If m is a term, then m*2^k is another term.
%C A334423 The first 3 primitive terms are 1, 21, 2231 and the 3 corresponding subsequences of such fixed points are,
%C A334423 -> m = 0 or m = 2^k, k>=0 (A131577),
%C A334423 -> m = 21 * 2^k, k>=0 (A175805),
%C A334423 -> m = 2231 * 2^k, k>=0 (2231, 4462, 9324, 18648, ...).
%F A334423 A257345(A004290(a(n))) = a(n).
%e A334423 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.
%e A334423 Successive operations for first primitive terms:
%e A334423 1 --> A004290(1) = 1_{10} --> 1_{2} = 1_{10},
%e A334423 21 --> A004290(21) = 10101_{10} --> 10101_{2} = 21_{10},
%e A334423 2231 --> A004290(2231) = 100010110111_{10} --> 100010110111_{2} = 2231_{10}.
%o A334423 (PARI) f(n) = {if( n==0, return (0)); my(m = n); while (vecmax(digits(m)) != 1, m+=n); m; } \\ A004290
%o A334423 isok(m) = fromdigits(digits(f(m), 10), 2) == m; \\ _Michel Marcus_, May 29 2020
%Y A334423 Cf. A004290, A079339, A257345.
%Y A334423 Subsequences: A131577, A175805.
%K A334423 nonn,base
%O A334423 1,3
%A A334423 _Bernard Schott_, May 25 2020