A020639 Lpf(n): least prime dividing n (when n > 1); a(1) = 1. Or, smallest prime factor of n, or smallest prime divisor of n.

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 3, 2, 7, 2, 79, 2, 3, 2, 83, 2, 5, 2, 3, 2, 89, 2, 7, 2, 3, 2, 5, 2, 97

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy
• nice
• core

Comments

Also, the largest number of distinct integers such that all their pairwise differences are coprime to n. - Max Alekseyev, Mar 17 2006

The unit 1 is not a prime number (although it has been considered so in the past). 1 is the empty product of prime numbers, thus 1 has no least prime factor. - Daniel Forgues, Jul 05 2011

a(n) = least m > 0 for which n! + m and n - m are not relatively prime. - Clark Kimberling, Jul 21 2012

For n > 1, a(n) = the smallest k > 1 that divides n. - Antti Karttunen, Feb 01 2014

For n > 1, records are at prime indices. - Zak Seidov, Apr 29 2015

The initials "lpf" might be mistaken for "largest prime factor" (A009190), using "spf" for "smallest prime factor" would avoid this. - M. F. Hasler, Jul 29 2015

n = 89 is the first index > 1 for which a(n) differs from the smallest k > 1 such that (2^k + n - 2)/k is an integer. - M. F. Hasler, Aug 11 2015

From Stanislav Sykora, Jul 29 2017: (Start)

For n > 1, a(n) is also the smallest k, 1 < k <= n, for which the binomial(n,k) is not divisible by n.

Proof: (A) When k and n are relatively prime then binomial(n,k) is divisible by n because k*binomial(n,k) = n*binomial(n-1,k-1). (B) When gcd(n,k) > 1, one of its prime factors is the smallest; let us denote it p, p <= k, and consider the binomial(n,p) = (1/p!)*Product_{i=0..p-1} (n-i). Since p is a divisor of n, it cannot be a divisor of any of the remaining numerator factors. It follows that, denoting as e the largest e > 0 such that p^e|n, the numerator is divisible by p^e but not by p^(e+1). Hence, the binomial is divisible by p^(e-1) but not by p^e and therefore not divisible by n. Applying (A), (B) to all considered values of k completes the proof. (End)

From Bob Selcoe, Oct 11 2017, edited by M. F. Hasler, Nov 06 2017: (Start)

a(n) = prime(j) when n == J (mod A002110(j)), n, j >= 1, where J is the set of numbers <= A002110(j) with smallest prime factor = prime(j). The number of terms in J is A005867(j-1). So:

a(n) = 2 when n == 0 (mod 2);

a(n) = 3 when n == 3 (mod 6);

a(n) = 5 when n == 5 or 25 (mod 30);

a(n) = 7 when n == 7, 49, 77, 91, 119, 133, 161 or 203 (mod 210);

etc. (End)

For n > 1, a(n) is the leftmost term, other than 0 or 1, in the n-th row of A127093. - Davis Smith, Mar 05 2019

References

• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.

Links

• Daniel Forgues, Table of n, a(n) for n = 1..100000 (terms 1..10000 from T. D. Noe)
• A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Copy included with the permission of the author.]
• OEIS Wiki, Least prime factor of n
• David Singmaster, Letter to N. J. A. Sloane, Oct 3 1982.
• Eric Weisstein's World of Mathematics, Least Prime Factor
• Index entries for "core" sequences

Crossrefs

Cf. A090368 (bisection).

Cf. A000040, A009190, A006530, A034684, A028233, A034699, A053585.

See also A032742, A055396, A068319, A088377, A007978, A053669, A117818.

Cf. A046669 (partial sums), A072486 (partial products).

Cf. A002110, A005867.

Cf. A127093.

Programs

Haskell

a020639 n = spf a000040_list where
  spf (p:ps) | n < p^2      = n
             | mod n p == 0 = p
             | otherwise    = spf ps
-- Reinhard Zumkeller, Jul 13 2011

Maple

A020639 := proc(n) if n = 1 then 1; else min(op(numtheory[factorset](n))) ; end if; end proc: seq(A020639(n),n=1..20) ; # R. J. Mathar, Oct 25 2010

Mathematica

f[n_]:=FactorInteger[n][[1,1]]; Join[{1}, Array[f,120,2]]  (* Robert G. Wilson v, Apr 06 2011 *)
Join[{1}, Table[If[EvenQ[n], 2, FactorInteger[n][[1,1]]], {n, 2, 120}]] (* Zak Seidov, Nov 17 2013 *)
Riffle[Join[{1},Table[FactorInteger[n][[1,1]],{n,3,101,2}]],2] (* Harvey P. Dale, Dec 16 2021 *)

PARI

A020639(n) = { vecmin(factor(n)[,1]) } \\ [Will yield an error for n = 1.] - R. J. Mathar, Mar 02 2012

PARI

A020639(n)=if(n>1, if(n>n=factor(n,0)[1,1], n, factor(n)[1,1]), 1) \\ Avoids complete factorization if possible. Often the smallest prime factor can be found quickly even if it is larger than primelimit. If factoring takes too long for large n, use debugging level >= 3 (\g3) to display the smallest factor as soon as it is found. - M. F. Hasler, Jul 29 2015

Python

from sympy import factorint
def a(n): return 1 if n == 1 else min(factorint(n))
print([a(n) for n in range(1, 98)]) # Michael S. Branicky, Dec 09 2021

Sage

def A020639_list(n) : return [1] + [prime_divisors(n)[0] for n in (2..n)]
A020639_list(97) # Peter Luschny, Jul 16 2012

Sage

[trial_division(n) for n in (1..100)] # Giuseppe Coppoletta, May 25 2016

Scheme

(define (A020639 n) (if (< n 2) n (let loop ((k 2)) (cond ((zero? (modulo n k)) k) (else (loop (+ 1 k))))))) ;; Antti Karttunen, Feb 01 2014

Formula

A014673(n) = a(A032742(n)); A115561(n) = a(A054576(n)). - Reinhard Zumkeller, Mar 10 2006

A028233(n) = a(n)^A067029(n). - Reinhard Zumkeller, May 13 2006

a(n) = A027746(n,1) = A027748(n,1). - Reinhard Zumkeller, Aug 27 2011

For n > 1: a(n) = A240694(n,2). - Reinhard Zumkeller, Apr 10 2014

a(n) = A000040(A055396(n)) = n / A032742(n). - Antti Karttunen, Mar 07 2017

a(n) has average order n/(2 log n) [Brouwer] - N. J. A. Sloane, Sep 03 2017

Extensions

Deleted wrong comment from M. Lagneau in 2012, following an observation by Gionata Neri. - M. F. Hasler, Aug 11 2015

Edited by M. F. Hasler, Nov 06 2017

Expanded definition to make this easier to find. - N. J. A. Sloane, Sep 21 2020