A053669 Smallest prime not dividing n.

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

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

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

Example

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

Maple

f:= proc(n) local p;
p:= 2;
while n mod p = 0 do p:= nextprime(p) od:
p
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
    while math.gcd(n, k) > 1: k += 1
    return k # Ely Golden, Nov 26 2020

Crossrefs

One more than A071222(n-1).

Cf. A000040, A020639, A053670, A053671, A053672, A053673, A053674, A055874, A079578, A087560, A096014, A235921, A236454, A249270, A257993 (the indices of these primes), A276086, A324895.

Sequence in context: A123556 A284017 A236454 * A342309 A112047 A112048

Adjacent sequences: A053666 A053667 A053668 * A053670 A053671 A053672

Keyword

nonn,nice,easy

Author

Henry Bottomley, Feb 15 2000

Extensions

More terms from Andrew Gacek (andrew(AT)dgi.net), Feb 21 2000 and James A. Sellers, Feb 22 2000

Entry revised by David W. Wilson, Nov 25 2006

Status

approved