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
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
KEYWORD
nonn
AUTHOR
STATUS
approved