The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A213636 Remainder when n is divided by its least nondivisor. 6
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
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
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
Sequence in context: A363057 A242481 A228287 * A192393 A184303 A218545
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 16 2012
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 29 12:57 EDT 2024. Contains 372940 sequences. (Running on oeis4.)