A291460 Even numbers m whose decimal expansion contains the decimal expansion of the greatest odd divisor of m as a substring.

16, 128, 384, 512, 1024, 1536, 1792, 2176, 2560, 2912, 3072, 5120, 7168, 8192, 9216, 11264, 13312, 15360, 15616, 16384, 17408, 19456, 21504, 23552, 25600, 27648, 28672, 29696, 31744, 33792, 35840, 37376, 37888, 39936, 41984, 43392, 57344, 66560, 90112, 98304

Offset: 1

Author

Bobby Jacobs, Aug 24 2017

Keywords

• base
• nonn

Comments

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?

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.

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

From Robert G. Wilson v, Aug 25 2017: (Start)

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

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

Examples

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1567 (terms < 10^20)

Crossrefs

Cf. A000265, A035057, A291555.

Programs

Mathematica

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 *)

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

Python

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

Extensions

More terms from Giovanni Resta, Aug 24 2017

Name edited by Felix Fröhlich, Aug 24 2017