Voozh

A053669

Smallest prime not dividing n.

200

2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Smallest prime coprime to n.

Smallest k >= 2 coprime to n.

a(#(p-1)) = a(A034386(p-1)) = p is the first appearance of prime p in sequence.

a(A005408(n)) = 2; for n > 2: a(n) = A112484(n,1). - Reinhard Zumkeller, Sep 23 2011

Average value is 2.920050977316134... = A249270. - Charles R Greathouse IV, Nov 02 2013

Differs from A236454, "smallest number not dividing n^2", for the first time at n=210, where a(210)=11 while A236454(210)=8. A235921 lists all n for which a(n) differs from A236454. - Antti Karttunen, Jan 26 2014

For k >= 0, a(A002110(k)) is the first occurrence of p = prime(k+1). Thereafter p occurs whenever A007947(n) = A002110(k). Thus every prime appears in this sequence infinitely many times. - David James Sycamore, Dec 04 2024

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, The American Mathematical Monthly, Vol. 126, No. 1 (2019), pp. 70-73; ResearchGate link.

James Grime and Brady Haran, 2.920050977316, Numberphile video (2020)

Igor Rivin, Geodesics with one self-intersection, and other stories, Advances in Mathematics, Vol. 231, No. 5 (2012), pp. 2391-2412; arXiv preprint, arXiv:0901.2543 [math.GT], 2009-2011.

FORMULA

a(n) = A071222(n-1)+1. [Because the right hand side computes the smallest k >= 2 such that gcd(n,k) = gcd(n-1,k-1) which is equal to the smallest k >= 2 coprime to n] - Antti Karttunen, Jan 26 2014

a(n) = 1 + Sum_{k=1..n}(floor((n^k)/k!)-floor(((n^k)-1)/k!)) = 2 + Sum_{k=1..n} A001223(k)*( floor(n/A002110(k))-floor((n-1)/A002110(k)) ). - Anthony Browne, May 11 2016

a(n!) = A151800(n). - Anthony Browne, May 11 2016

a(2k+1) = 2. - Bernard Schott, Jun 03 2019

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A249270. - Amiram Eldar, Oct 29 2020

a(n) = A000040(A257993(n)) = A020639(A276086(n)) = A276086(n) / A324895(n). - Antti Karttunen, Apr 24 2022

a(n) << log n. For every e > 0, there is some N such that for all n > N, a(n) < (1 + e)*log n. - Charles R Greathouse IV, Dec 03 2022

A007947(n) = A002110(k) ==> a(n) = prime(k+1). - David James Sycamore, Dec 04 2024

EXAMPLE

a(60) = 7, since all primes smaller than 7 divide 60 but 7 does not.

a(90) = a(120) = a(150) = a(180) = 7 because 90,120,150,180 all have same squarefree kernel = 30 = A002110(3), and 7 is the smallest prime which does not divide 30. - David James Sycamore, Dec 04 2024

MAPLE

f:= proc(n) local p;

p:= 2;

while n mod p = 0 do p:= nextprime(p) od:

end proc:

map(f, [$1..100]); # Robert Israel, May 18 2016

MATHEMATICA

Table[k := 1; While[Not[GCD[n, Prime[k]] == 1], k++ ]; Prime[k], {n, 1, 60}] (* Stefan Steinerberger, Apr 01 2006 *)

With[{prs=Prime[Range[10]]}, Flatten[Table[Select[prs, !Divisible[ n, #]&, 1], {n, 110}]]] (* Harvey P. Dale, May 03 2012 *)

PROG

(Haskell)

a053669 n = head $ dropWhile ((== 0) . (mod n)) a000040_list

-- Reinhard Zumkeller, Nov 11 2012

(PARI) a(n)=forprime(p=2, , if(n%p, return(p))) \\ Charles R Greathouse IV, Nov 20 2012

(Scheme) (define (A053669 n) (let loop ((i 1)) (cond ((zero? (modulo n (A000040 i))) (loop (+ i 1))) (else (A000040 i))))) ;; Antti Karttunen, Jan 26 2014

(Python)

from sympy import nextprime

def a(n):

p = 2

while True:

if n%p: return p

else: p=nextprime(p) # Indranil Ghosh, May 12 2017

(Python)

# using standard library functions only

import math

def a(n):

k = 2