A230399 -id:A230399 - cp's OEIS frontend

A334817 Distinct values of A230399(n) - n.

1, 6, 20, 24, 30, 48, 60, 112, 160, 168, 180, 336, 360, 480, 720, 1008, 1080, 1260, 1440, 2100, 2400, 2520, 2880, 3960, 4200, 4680, 5040, 5280, 6720, 7560, 8400, 9240, 12600, 12960, 13440, 13860, 16800, 18480, 20160, 23100, 25200, 27720, 30240, 32760, 37800, 40320, 41580

Offset: 1

David A. Corneth, May 12 2020

nonn

For many terms there are lots of solutions to the equation a(n) = A230399(m) - m. For example there are 5 solutions to 360 = A230399(m) - m, and 5068 solutions to 49008960 = A230399(m) - m.

			24 is in the sequence as A230399(2) - 2 = 26-2 = 24.

Cf. A230374, A230399.

A072528 Table T(n,k) read by rows, giving number of occurrences of the remainder k when n is divided by i=1,2,3,...,n.

Original entry on oeis.org

1, 2, 2, 1, 3, 1, 2, 2, 1, 4, 1, 1, 2, 3, 1, 1, 4, 1, 2, 1, 3, 3, 1, 1, 1, 4, 2, 2, 1, 1, 2, 3, 2, 2, 1, 1, 6, 1, 2, 1, 1, 1, 2, 5, 1, 2, 1, 1, 1, 4, 1, 4, 1, 2, 1, 1, 4, 3, 1, 3, 1, 1, 1, 1, 5, 3, 2, 1, 2, 1, 1, 1, 2, 4, 3, 2, 1, 2, 1, 1, 1, 6, 1, 3, 2, 2, 1, 1, 1, 1, 2, 5, 1, 3, 2, 2, 1, 1, 1, 1, 6, 1, 4, 1, 2

Offset: 1

Amarnath Murthy, Aug 01 2002

The n-th row adds to n.

			The table begins
1
2
2 1
3 1
2 2 1
4 1 1
2 3 1 1
4 1 2 1

Cf. A023645 for T(n, 2) and A072527 for T(n, 3).

Cf. A230374, A230399.

Let a(m) be the m-th term in the sequence. Then m=f(n)+k where f(1)=1 and f(n+1)=f(n)+floor((n+1)/2). n is the number being divided by the various i's and k is the remainder under consideration. f(n) has the generating function F(x)= (x(1+2x^2-2x^3))/((1-x)^2(1+x^2)) - Bruce Corrigan (scentman(AT)myfamily.com), Oct 22 2002

G.f. for k-th column: Sum_{m>0} x^((k+1)*m+k)/(1-x^m). - Vladeta Jovovic, Dec 16 2002

Edited by Bruce Corrigan (scentman(AT)myfamily.com), Oct 22 2002

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 25 2003

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

Original entry on oeis.org

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

Vladimir Letsko, Oct 17 2013

nonn

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

			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.

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

Cf. A002182, A230399.

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:

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 *)

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

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A334817 Distinct values of A230399(n) - n.

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A072528 Table T(n,k) read by rows, giving number of occurrences of the remainder k when n is divided by i=1,2,3,...,n.

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

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