A117913 -id:A117913 - cp's OEIS frontend

A115569 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.

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 24, 36, 48, 124, 126, 128, 132, 135, 162, 168, 175, 184, 216, 248, 264, 312, 315, 324, 384, 396, 412, 432, 612, 624, 648, 672, 728, 735, 784, 816, 824, 864, 936, 1236, 1248, 1296, 1326, 1362, 1368, 1395, 1632, 1692, 1764, 1824

Offset: 1

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

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

			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.

Rick L. Shepherd, List of all terms

Cf. A034838, A034709, A063527.

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

with(combinat):
f:= l-> parse(cat(l[])):
T:= n-> sort(map(f, select(l-> andmap(x-> irem(f(l), x)=0, l),
         map(p-> permute(p)[], choose([$1..9], n)))))[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Jul 31 2022

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

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

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

def ok(n):
    s = str(n)
    if "0" in s or len(set(s)) < len(s): return False
    return all(n%int(d) == 0 for d in s)
afull = [k for k in range(9867313) if ok(k)]
print(afull[:55]) # Michael S. Branicky, Jul 31 2022

The full list of terms was sent in by Rick L. Shepherd (see link) and also by Sébastien Dumortier, Apr 04 2006

cp's OEIS Frontend

A115569 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Extensions

A117912 Lynch-Bell numbers whose digits are all even.

Views

Author

Keywords

Crossrefs