

A291460


Even numbers n such that the decimal expansion of n contains the decimal expansion of the greatest odd divisor of n as a substring.


2



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
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OFFSET

1,1


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*k5) for k>0.  Giovanni Resta, Aug 25 2017
Trivially all powers of two which contain the digit 1 are members, see A035057.  Robert G. Wilson v, Aug 25 2017
Number of terms less than 10^k: 0, 1, 4, 15, 40, 57, 76, 108, 146, 194, 258, 336, 447, etc.  Robert G. Wilson v, Aug 25 2017


LINKS

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


EXAMPLE

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.


MAPLE

with(numtheory): 2*select(n>searchtext("".(max(op(select(type, divisors(2*n), odd)))), "".(2*n))>0, [$1..50000]); # Paolo P. Lava, Sep 05 2018


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 *)


PROG

(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


CROSSREFS

Cf. A000265, A035057, A291555.
Sequence in context: A133111 A268998 A253319 * A004017 A167471 A153115
Adjacent sequences: A291457 A291458 A291459 * A291461 A291462 A291463


KEYWORD

base,nonn


AUTHOR

Bobby Jacobs, Aug 24 2017


EXTENSIONS

More terms from Giovanni Resta, Aug 24 2017
Name edited by Felix FrÃ¶hlich, Aug 24 2017


STATUS

approved



