A230374 - OEIS

%I #19 Jul 02 2019 19:36:37

%S 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,

%T 72,78,80,84,90,96,100,102,104,105,108,112,120,126,132,138,140,144,

%U 150,156,160,168,176,180,192,198,200,204,208,210,216,224,228,234,240,252

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

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

%H Ivan Neretin, <a href="/A230374/b230374.txt">Table of n, a(n) for n = 1..1000</a>

%e 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.

%e 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.

%p rem0:=proc(n) local r,n1,i,mx,f,R;

%p n1:=`if`(n mod 2 = 0, n/2-1,(n-1)/2);

%p R:=Array(0..n1,fill=1):if n mod 2 = 0 then R[0]:=2 fi:

%p for i to n1 do r:=n mod i: R[r]:=R[r]+1 od:

%p mx:=R[0]:f:=true:

%p for i to n1 do

%p if R[i]>= mx then f:=false:break fi od:

%p f; end;

%p for n do if maxrem(n) then print(n) fi od:

%t Select[Range[256], (r = (Transpose@Tally@Mod[#, Range@#])[[2]])[[1]] > Max@Rest@r &] (* _Ivan Neretin_, Nov 13 2016 *)

%t 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 *)

%o (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

%Y Cf. A002182, A230399.

%K nonn

%O 1,2

%A _Vladimir Letsko_, Oct 17 2013