A291460 - OEIS

%I #65 Mar 11 2024 01:50:49

%S 16,128,384,512,1024,1536,1792,2176,2560,2912,3072,5120,7168,8192,

%T 9216,11264,13312,15360,15616,16384,17408,19456,21504,23552,25600,

%U 27648,28672,29696,31744,33792,35840,37376,37888,39936,41984,43392,57344,66560,90112,98304

%N Even numbers m whose decimal expansion contains the decimal expansion of the greatest odd divisor of m as a substring.

%C 43392 and 443392 are both in this sequence because 43392 = 2^7*339 and 443392 = 2^10*433. Removing the first digit of 443392 gives 43392. Are there any other numbers in this sequence such that removing the first digit gives another number in the sequence?

%C Every number of the form 29090...90912 is in this sequence because 2912 = 2^5*91, 290912 = 2^5*9091, 29090912 = 2^5*909091, and so on.

%C Sequence is infinite since it contains infinite subsequences, like the numbers of the form 7*2^(20*k-5) for k > 0. - _Giovanni Resta_, Aug 25 2017

%C From _Robert G. Wilson v_, Aug 25 2017: (Start)

%C Trivially all powers of two which contain the digit 1 are terms, see A035057.

%C Number of terms less than 10^k: 0, 1, 4, 15, 40, 57, 76, 108, 146, 194, 258, 336, 447, etc. (End)

%H Giovanni Resta, <a href="/A291460/b291460.txt">Table of n, a(n) for n = 1..1567</a> (terms < 10^20)

%e The greatest odd divisor of 1792 = 2^8*7 is 7, which is in 1(7)92. Therefore, 1792 is in this sequence.

%e 2^10*x = 1024*x contains x as a substring for all x from 1 to 41. Therefore, 1024*x is in this sequence for all odd x from 1 to 41.

%t inQ[n_] := StringPosition[ToString[n], ToString[n/2^IntegerExponent[n, 2]]] != {}; Select[2 Range[50000], inQ] (* _Giovanni Resta_, Aug 24 2017 and slightly modified by _Robert G. Wilson v_, Aug 25 2017 *)

%o (PARI) is(n)=if(n%2, return(0)); my(r=n>>valuation(n, 2), m=Mod(r, 10^#digits(r))); while(n>=r, if(n==m, return(1)); n\=10); 0 \\ _Charles R Greathouse IV_, Aug 26 2017

%o (Python)

%o A291460_list = [2*x for x in range(1,10**6) if str(int(bin(x).rstrip('0'),2)) in str(2*x)] # _Chai Wah Wu_, Aug 31 2017

%Y Cf. A000265, A035057, A291555.

%K base,nonn

%O 1,1

%A _Bobby Jacobs_, Aug 24 2017

%E More terms from _Giovanni Resta_, Aug 24 2017

%E Name edited by _Felix Fröhlich_, Aug 24 2017