A230374 -id:A230374 - cp's OEIS frontend

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.

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

A230399 Smallest k such that when k is divided by all numbers <= k the remainder n occurs most often.

Original entry on oeis.org

1, 7, 26, 23, 34, 53, 118, 167, 188, 69, 178, 179, 372, 349, 374, 375, 376, 377, 498, 499, 356, 501, 502, 503, 1284, 1285, 746, 747, 748, 749, 1038, 1039, 1112, 753, 754, 755, 2136, 2137, 2138, 2139, 2140, 2141, 2562, 2443, 1484, 2445, 1486, 1487, 2568, 2569, 2570, 2571, 2572, 2573, 2934, 2575

Offset: 0

Vladimir Letsko, Oct 18 2013

Number of row in A072528 where it happens first that the n-th column contains the maximum value in the row. - Ralf Stephan, Oct 21 2013
Any nonnegative integer n is the most common remainder during dividing m by all numbers not exceeding m for infinitely many different positive integers m.
Ties for most common are not allowed. Thus a(1) is not 5, although 0 and 1 both occur twice among 5 mod 1 to 5 mod 5. - Robert Israel, May 10 2020
Vladimir Letsko (see Letsko Link) asks if a(2013) exists (paraphrasing his question) without asking its value. Indeed, for m = 526173 we have T(m, 2013) = 68 where 68 is the largest value of T(526173, k) for k >= 0 and T as defined in A072528. - David A. Corneth, May 12 2020

			a(1) = 7 because
(1) 7 mod 2 = 7 mod 3 = 7 mod 6 = 1 and the other remainders occur fewer times;
(2) 7 is the least number k for which r = k mod b yields the remainder r=1 for more bases b < k than any other remainder.

David A. Corneth, Table of n, a(n) for n = 0..10080 (first 468 terms from Robert Israel)
Vladimir Letsko, Mathematical Marathon, Problem 181 (in Russian).

Cf. A230374, A002182, A072528.

maxrem:=proc(n) local r,n1,i,mx,M,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]:M:={0}:
for i to n1 do
if R[i]> mx then mx:=R[i]:M:={i} elif mx=R[i] then M:=M union {i} fi od:
M; end;
Rs:={0}:S:=[[0,1]]:for n to 6000 do r:=maxrem(n):if nops(r)=1 then r:=op(r):
if not member(r,Rs) then Rs:=Rs union {r}:S:=[op(S),[r,n]] fi fi od:
S:=sort(S);
T:=[]:for i to nops(S) do if S[i,1]=i-1 then T:=[op(T),S[i,2]] else break fi od:T;

a[n_] := Module[{k, rems}, For[k = 1, True, k++, rems = SortBy[Tally[Mod[k, Range[k]]], Last]; If[rems[[-1, 1]] == n && rems[[-1, 2]] != rems[[-2, 2]], Print[n, " ", k]; Return[k]]]];
a[0] = 1;
a /@ Range[0, 100] (* Jean-François Alcover, Jun 18 2020 *)

record(n)=v=vector(n+1); for(d=1,n, t=(n%d)+1; v[t]=v[t]+1); m=0; p=0; for(i=1,n,if(v[i]>m, m=v[i]; p=i));p
for(n=1,100, for(j=1,10^6, if(record(j)==n, print1(j,", "); break))) \\ Ralf Stephan, Oct 21 2013

A334817 Distinct values of A230399(n) - n.

Original entry on oeis.org

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.

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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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A230399 Smallest k such that when k is divided by all numbers <= k the remainder n occurs most often.

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A334817 Distinct values of A230399(n) - n.

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