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A213636 Remainder when n is divided by its least nondivisor. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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 - 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 map(a, range(1, 101)) # Indranil Ghosh, Jul 28 2017

CROSSREFS

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

Sequence in context: A307614 A242481 A228287 * A192393 A184303 A218545

Adjacent sequences:  A213633 A213634 A213635 * A213637 A213638 A213639

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 16 2012

STATUS

approved

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Last modified May 30 15:21 EDT 2020. Contains 334726 sequences. (Running on oeis4.)