A007978 Least non-divisor of n.

2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3

Offset

1,1

Comments

Least k >= 2 such that sigma(n) divides sigma(n*k), where sigma is A000203. - Benoit Cloitre, Dec 01 2002

Contains all and only the prime powers p^k, k > 0. The first occurrence of p^k is at A003418(p^k-1); so new records occur at indices in A051451. - Franklin T. Adams-Watters, Jun 13 2011

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Bakir Farhi, On the average asymptotic behavior of a certain type of sequences of integers, Integers, Vol. 9 (2009), pp. 555-567.

Formula

a(n) = A053669(n) + A061853(n) = A055874(n) + 1. - Henry Bottomley, May 10 2001

G.f.: sum(k >= 2, -k*(x^A003418(k) - x^A003418(k-1))/((x^A003418(k) - 1)*(x^A003418(k-1) - 1))). - Robert Israel, Sep 02 2014

From Alonso del Arte, Sep 23 2017: (Start)

a(n) < n for all n > 2.

a(2n + 1) = 2, a(2n) >= 3.

a(2^k) = 3 for k > 0.

a(n!) = prime(pi(n) + 1) for n >= 0, except for a(3!) = 4. (End)

Asymptotic mean: lim_{n->oo} Sum_{k=1..n} a(k) = 1 + A064859 (Farhi, 2009). - Amiram Eldar, Jun 29 2021

Maple

a:= proc(n) local k;
for k from 2 while n mod k = 0 do od:
k
end proc:
seq(a(n), n=1..100); # Robert Israel, Sep 02 2014

Mathematica

Table[k := 1; While[Mod[n, k] == 0, k++]; k, {n, 2000}]  (* Clark Kimberling, Jun 16 2012 *)
Join[{2, 3}, Table[Complement[Range[n], Divisors[n]][[1]], {n, 3, 100}]] (* Alonso del Arte, Sep 23 2017 *)

Prog

(Haskell)
import Data.List ((\\))
a007978 = head . ([1..] \\) . a027750_row
-- Reinhard Zumkeller, May 10 2014
(PARI) a(n) = {my(k=2); while(!(n % k), k++); k; } \\ Michel Marcus, Sep 25 2017
(Python)
def a(n):
    k = 2
    while not n%k: k += 1
    return k
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 09 2022
(Python)
def A007978(n): return next(filter(lambda d:n%d, range(2, n))) if n>2 else n+1 # Chai Wah Wu, Feb 22 2023

Crossrefs

Cf. A003418, A051451, A055874, A053669, A061853, A027750, A064859.

Cf. also A266620 (least non-divisor of n!).

Sequence in context: A228829 A341982 A337686 * A245575 A333600 A339625

Adjacent sequences: A007975 A007976 A007977 * A007979 A007980 A007981

Keyword

nonn

Author

Jeffrey Shallit

Status

approved