A230374 The numbers n such that during dividing n by all positive integers not exceeding n the remainder 0 occurs most often.

1, 2, 3, 4, 6, 8, 10, 12, 15, 16, 18, 20, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 102, 104, 105, 108, 112, 120, 126, 132, 138, 140, 144, 150, 156, 160, 168, 176, 180, 192, 198, 200, 204, 208, 210, 216, 224, 228, 234, 240, 252

Offset: 1

Author

Vladimir Letsko, Oct 17 2013

Keywords

• nonn

Comments

A natural generalization of highly composite numbers (A002182), which is a subsequence of this sequence.

Examples

			8 is in the sequence because remainder 0 occurs 4 times during division 8 by 1, 2, 3, 4, 5, 6, 7, 8, that is more than other remainders.
9 is not in the sequence because both remainders 0 and 1 occur 3 times during division 9 by 1, 2, 3, 4, 5, 6, 7, 8, 9.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002182, A230399.

Programs

Maple

rem0:=proc(n) local r,n1,i,mx,f,R;
n1:=`if`(n mod 2 = 0, n/2-1,(n-1)/2);
R:=Array(0..n1,fill=1):if n mod 2 = 0 then R[0]:=2 fi:
for i to n1 do r:=n mod i: R[r]:=R[r]+1 od:
mx:=R[0]:f:=true:
for i to n1 do
if R[i]>= mx then f:=false:break fi od:
f; end;
for n do if maxrem(n) then print(n) fi od:

Mathematica

Select[Range[256], (r = (Transpose@Tally@Mod[#, Range@#])[[2]])[[1]] > Max@Rest@r &] (* Ivan Neretin, Nov 13 2016 *)
zmoQ[n_] := Module[{r = Sort[Tally[Mod[n, Range[n]]]], mx}, mx = Select[r, #[[2]] == Max[r[[All, 2]]] &]; Length[mx] == 1 && mx[[1, 1]] == 0]; Select[ Range[300],zmoQ] (* Harvey P. Dale, Jul 02 2019 *)

PARI

is(n)=v=vector(n+1);for(d=1,n,t=(n%d)+1;v[t]=v[t]+1);nd=v[1];for(i=2,n,if(v[i]>=nd,return(0)));1 \\ Ralf Stephan, Oct 21 2013