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

1,2

Comments

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A002182, A230399.

Sequence in context: A064375 A037229 A351740 * A007183 A067783 A062418

Adjacent sequences: A230371 A230372 A230373 * A230375 A230376 A230377

Keyword

nonn

Author

Vladimir Letsko, Oct 17 2013

Status

approved