A213636 Remainder when n is divided by its least nondivisor.

1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1

Offset

1,2

Comments

Experimentation suggests that every positive integer occurs in this sequence and that

2 occurs only in even numbered positions,

3 occurs in only in positions that are multiples of 12,

4 occurs only in positions that are multiples of 12,

5 occurs only in positions that are multiples of 60,

6 occurs only in positions that are multiples of 60,

7 occurs only in positions that are multiples of 2520, etc.

See A213637 for positions of 1 and A213638 for positions of 2.

From Robert Israel, Jul 28 2017: (Start)

Given any positive number m, let q be a prime > m and r = A003418(q-1). Then a(n) = m if n == m (mod q) and n == 0 (mod r). By the Chinese Remainder Theorem, such n exists.

On the other hand, if a(n) = m, we must have A007978(n) > m, and then n must be divisible by A003418(q-1) where q = A007978(n) is a member of A000961 greater than m. Moreover, if q=p^j with j>1, n is divisible by p^(j-1) so m must be divisible by p^(j-1). Thus:

For m=2, A003418(2)=2.

For m=3, A007978(n) can't be 4 because m is odd, so A007978(n)>= 5 and n must be divisible by A003418(4)=12.

For m=4, A003418(4)=12.

For m=5 or 6, A003418(6)=60.

For m=7, A007978(n) can't be 8 because m is odd, and can't be 9 because m is not divisible by 3, so n must be divisible by A003418(10)=2520. (End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(n) = n - A213635(n).

a(n) = n - m(n)*floor(n/m(n)), where m(n) = A007978(n).

Example

a(10) = 10-3*[10/3] = 1.

Maple

f:= proc(n) local k;
  for k from 2 do if n mod k <> 0 then return n mod k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Jul 27 2017

Mathematica

y=120; z=2000;
t = Table[k := 1; While[Mod[n, k] == 0, k++];
   k, {n, 1, z}]  (*A007978*)
Table[Floor[n/t[[n]]], {n, 1, y}] (*A213633*)
Table[n - Floor[n/t[[n]]], {n, 1, y}] (*A213634*)
Table[t[[n]]*Floor[n/t[[n]]], {n, 1, y}] (*A213635*)
t1 = Table[n - t[[n]]*Floor[n/t[[n]]],
   {n, 1, z}] (* A213636 *)
Flatten[Position[t1, 1]] (* A213637 *)
Flatten[Position[t1, 2]] (* A213638 *)
rem[n_]:=Module[{lnd=First[Complement[Range[n], Divisors[n]]]}, Mod[n, lnd]]; Join[{1, 2}, Array[rem, 100, 3]] (* Harvey P. Dale, Mar 26 2013 *)
Table[Mod[n, SelectFirst[Range[n + 1], ! Divisible[n, #] &]], {n, 105}] (* Michael De Vlieger, Jul 29 2017 *)

Prog

(Scheme) (define (A213636 n) (modulo n (A007978 n))) ;; Antti Karttunen, Jul 27 2017
(Python)
def a(n):
    k=2
    while n%k==0: k+=1
    return n%k
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 28 2017
(Python)
def A213636(n): return next(filter(None, (n%d for d in range(2, n)))) if n>2 else n # Chai Wah Wu, Feb 22 2023

Crossrefs

Cf. A000961, A003418, A007978, A213633, A213635, A213637, A213638.

Sequence in context: A363057 A242481 A228287 * A192393 A184303 A218545

Adjacent sequences: A213633 A213634 A213635 * A213637 A213638 A213639

Keyword

nonn

Author

Clark Kimberling, Jun 16 2012

Status

approved