A115569 - OEIS

%I #38 Sep 03 2024 01:28:37

%S 1,2,3,4,5,6,7,8,9,12,15,24,36,48,124,126,128,132,135,162,168,175,184,

%T 216,248,264,312,315,324,384,396,412,432,612,624,648,672,728,735,784,

%U 816,824,864,936,1236,1248,1296,1326,1362,1368,1395,1632,1692,1764,1824

%N Lynch-Bell numbers: numbers n such that the digits are all different (and do not include 0) and n is divisible by each of its individual digits.

%C This is a subset of some of the related sequences listed below. Stephen Lynch and Andrew Bell are Brisbane surgeons who contributed to the identification of this sequence.

%C There are 548 Lynch-Bell numbers. A117911 gives the number of n-digit ones. The digit 5 cannot appear in Lynch-Bell numbers containing an even digit; 5 must be the units digit when it appears. The 7-digit Lynch-Bell numbers are 105 permutations of 1289736 (the smallest such). - _Rick L. Shepherd_, Apr 01 2006

%C Can be seen/read as a table with row lengths A117911 (rows r > 7 have zero length). - _M. F. Hasler_, Jan 31 2016

%H Rick L. Shepherd, <a href="/A115569/b115569.txt">List of all terms</a>

%e 384/3 = 128, 384/8 = 48, 384/4 = 96. Thus 384 is Lynch-Bell as it is a multiple of each of its three distinct digits.

%p with(combinat):

%p f:= l-> parse(cat(l[])):

%p T:= n-> sort(map(f, select(l-> andmap(x-> irem(f(l), x)=0, l),

%p          map(p-> permute(p)[], choose([$1..9], n)))))[]:

%p seq(T(n), n=1..7);  # _Alois P. Heinz_, Jul 31 2022

%t Reap[For[n = 1, n < 10^7, n++, id = IntegerDigits[n]; If[FreeQ[id, 0] && Length[id] == Length[Union[id]] && And @@ (Divisible[n, #]& /@ id), Print[n]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Nov 26 2013 *)

%t bnQ[n_]:=Max[DigitCount[n]]==1&&FreeQ[IntegerDigits[n],0]&&Union[Divisible[n,IntegerDigits[ n]]]=={True}; Select[Range[2000],lbnQ] (* _Harvey P. Dale_, Jun 02 2023 *)

%t Cases[Union @@ ((FromDigits@#&/@Flatten[Permutations@# & /@ Subsets[Range@9, {#}], 1])&/@ Range@9), _?(DeleteDuplicates[Divisible[#, IntegerDigits@#]] == {True} &)] (* _Hans Rudolf Widmer_, Aug 27 2024 *)

%o (PARI) A115569_row(n)={if(n,my(u=vectorv(n,i,10^i)\10,S=List(),M);forvec(v=vector(n,i,[1,9]),(M=lcm(v))%10==0||normlp(v,1)%3^valuation(M,3)||for(k=1,n!,vecextract(v,numtoperm(n,k))*u%M ||listput(S,vecextract(v,numtoperm(n,k))*u)),2);Set(S),concat(apply(A115569_row,[1..7])))} \\ Return terms of length n if given, else the vector of all terms. The checks M%10 and |v| % 3^v(...) are not needed but reduce CPU time by 97%. - _M. F. Hasler_, Jan 31 2016

%o (PARI) A115569(n)=n>9&&for(r=2,7,(n-=#t=A115569_row(r))>9||return(t[n-9+#t]));n \\ _M. F. Hasler_, Jan 31 2016

%o (Python)

%o def ok(n):

%o     s = str(n)

%o     if "0" in s or len(set(s)) < len(s): return False

%o     return all(n%int(d) == 0 for d in s)

%o afull = [k for k in range(9867313) if ok(k)]

%o print(afull[:55]) # _Michael S. Branicky_, Jul 31 2022

%Y Cf. A034838, A034709, A063527.

%Y Cf. A117911, A117912 (have even digits only), A117913 (have odd digits only), A010784.

%K base,easy,nonn,fini,full

%O 1,2

%A Mike Smith (mtm_king(AT)yahoo.com), Mar 10 2006; also submitted by Andy Edwards (AndynGen(AT)aol.com), Mar 20 2006

%E The full list of terms was sent in by Rick L. Shepherd (see link) and also by _Sébastien Dumortier_, Apr 04 2006