A265730 - OEIS

%I #17 Dec 23 2024 14:53:44

%S 1,2,3,4,5,6,7,8,9,11,16,17,27,47,52,57,78,87,117,131,151,157,158,161,

%T 171,177,187,191,247,257,277,417,447,457,487,517,547,557,558,587,592,

%U 717,747,757,787,817,857,887,1131,1147,1151,1157,1161,1167,1171,1177,1181,1191,1221,1247,1251,1257,1261,1287,1331,1361,1371,1417

%N Zeroless numbers n such that the decimal expansion of a/b contains each digit of n, for all a,b > 0 such that concatenate(a,b) = n.

%C Termed "1951 numbers" by Eric Angelini, cf. LINKS.

%H E. Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-December/015839.html">1951-numbers</a>, SeqFan list, Dec 14, 2015.

%e The single-digit numbers 1,...,9 are in the sequence because there is no possible decomposition a,b.

%e The number 1951 is in the sequence because 195/1 = 195.0, 19/51 = 0.3725490196... and 1/951 = 0.0010515247108307045215562565720294... all have each of the digits '1', '5' and '9' the number 1951 is made of.

%t fQ[n_] := Block[{d = IntegerDigits@ n, w}, w = Map[Union@ Flatten@ # &, First /@ RealDigits@ Map[FromDigits@ Take[d, #]/FromDigits@ Take[d, -Length@ d + #] &, Reverse@ Range[Length@ d - 1]], {1}]; And @@ Function[k, AllTrue[d, MemberQ[k, #] &]] /@ w]; Select[Select[Range@ 1500, Last@ DigitCount@ # == 0 &], fQ] (* _Michael De Vlieger_, Dec 14 2015 *)

%o (PARI) is(n,d=Set(digits(n)),p=n+#Str(n))={vecmin(d)&&!for(i=10,n,setminus(d,Set(digits(n\i*10^p\(n%i))))&&return;i=i*10-1)}

%Y Cf. A051626.

%K nonn,base

%O 1,2

%A _Eric Angelini_ and _M. F. Hasler_, Dec 14 2015