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Non-"Ding!Bong!" numbers: positive numbers which are not a positive linear combination of 5's and 7's.
2

%I #19 Mar 06 2019 19:35:15

%S 1,2,3,4,6,8,9,11,13,16,18,23

%N Non-"Ding!Bong!" numbers: positive numbers which are not a positive linear combination of 5's and 7's.

%C From the "Ding!Bong!" game: list the natural numbers replacing 5 by Ding! and 7 by Bong! All numbers except those listed in the sequence are combinations of Dings or of Bongs (this includes all numbers >23).

%C Positive numbers not of the form 5x + 7y with nonnegative x and y.

%H Gianni A. Sarcone and Marie-Jo Waeber, <a href="http://www.archimedes-lab.org/numbers/Num1_69.html#dingbong">Can you count in 'ding-bong'?</a>.

%e 1 2 3 4 Ding! 6 Bong! 8 9 Ding-Ding! 11 Ding-Bong! 13 Bong-Bong! Ding-Ding-Ding! 16 Ding-Ding-Bong! 18 Ding-Bong-Bong! Ding-Ding-Ding-Ding! Bong-Bong-Bong! Ding-Ding-Ding-Bong! 23 Ding-Ding-Bong-Bong! Ding-Ding-Ding-Ding-Ding! Ding-Bong-Bong-Bong! Ding-Ding-Ding-Ding-Bong! Bong-Bong-Bong-Bong! Ding-Ding-Ding-Bong-Bong! Ding-Ding-Ding-Ding-Ding-Ding!

%t Position[Table[FrobeniusSolve[{5,7},n],{n,23}]/.{}->r,r]//Flatten (* _Harvey P. Dale_, Mar 06 2019 *)

%Y Cf. A000027, A102705.

%K easy,fini,full,nonn

%O 1,2

%A _Alexandre Wajnberg_, Feb 04 2005

%E Corrected by _Zak Seidov_, Oct 22 2011

%E Entry revised by _N. J. A. Sloane_, Mar 06 2019