OFFSET
1,2
COMMENTS
Also the numbers which are incrementally largest values of A002034. - validated by Franklin T. Adams-Watters, Jul 13 2012
Also numbers m such that m, phi(m) and tau(m) form an integer triangle, where phi=A000010 is the totient and tau=A000005 the number of divisors (see also A084820). - Reinhard Zumkeller, Jun 04 2003
Terms > 1 are n such that n does not divide (n-1)!. - Benoit Cloitre, Nov 12 2003
Terms > 1 are the sum of their prime factors; 4 (= 2+2) is the only such composite number. - Stuart Orford (sjorford(AT)yahoo.co.uk), Aug 04 2005
From Jonathan Vos Post, Aug 23 2010, Robert G. Wilson v, Aug 25 2010, proof by D. S. McNeil, Aug 29 2010: (Start)
Theorem: for a multiset of m >= 2 integers a_i, each a_i >= 2, Product_{i=1..m} a_i >= Sum_{i=1..m} a_i, with equality only at (a_1,a_2) = (2,2).
Lemma: For integers x,y >= 2, if x > 2 or y > 2, x*y > x + y. This follows from distributing (x-1)*(y-1) > 1.
[Proof of the theorem by induction on m:
first consider m=2. We have equality at (2,2) and for any product(a_i) > 4 there is some a_i > 2, so the lemma gives a_1*a_2 > a_1+a_2.
Then the induction m->m+1: Product_{i=1..m+1} a_i = a_(m+1)*Product_{i=1..m} a_i >= a_(m+1) * Sum_{i=1..m} a_i.
Since a_(m+1) >= 2 and the sum >= 4, the lemma applies, and we find a_(m+1) * Sum+{i=1..m} a_i > a_(m+1) + Sum_{i=1..m} a_i = Sum_{i=1..m+1} a_i and thus Product_{i=1..m+1} a_i > Sum_{i=1..m+1} a_i, QED.]
For composite n > 4, applying the theorem to the multiset of prime factors with multiplicity yields n > sopfr(n), so there are no composite numbers greater than 4 such that they divide sopfr(n).
(End)
Numbers k such that the k-th Fibonacci number is relatively prime to all smaller Fibonacci numbers. - Charles R Greathouse IV, Jul 13 2012
Numbers k such that (-1)^k*floor(d(k)*(-1)^k/2) = 1, where d(k) is the number of divisors of k. - Wesley Ivan Hurt, Oct 11 2013
Also, union of odd primes (A065091) and the divisors of 4. Also, union of A008578 and 4. - Omar E. Pol, Nov 04 2013
Numbers k such that sigma(k!) is divisible by sigma((k-1)!). - Altug Alkan, Jul 18 2016
LINKS
J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function
Eric Weisstein's World of Mathematics, Sum of Prime Factors
FORMULA
A141295(a(n)) = a(n). - Reinhard Zumkeller, Jun 23 2008
A018194(a(n)) = 1. - Reinhard Zumkeller, Mar 09 2012
A240471(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2014
MAPLE
A046022:=n-> `if`((-1)^n*floor(numtheory[tau](n)*(-1)^n/2) = 1, n, NULL); seq(A046022(j), j=1..260); # Wesley Ivan Hurt, Oct 11 2013
MATHEMATICA
max = 0; a = {}; Do[m = FactorInteger[n]; w = Sum[m[[k]][[1]]*m[[k]][[2]], {k, 1, Length[m]}]; If[w > max, AppendTo[a, w]; max = w], {n, 1, 1000}]; a (* Artur Jasinski, Apr 06 2008 *)
PROG
(Haskell)
a046022 n = a046022_list !! (n-1)
a046022_list = [1..4] ++ drop 2 a000040_list
-- Reinhard Zumkeller, Apr 06 2014
(PARI) a(n)=if(n<6, n, prime(n-2)) \\ Charles R Greathouse IV, Apr 28 2015
(Python)
from sympy import prime
def A046022(n): return prime(n-2) if n>4 else n # Chai Wah Wu, Oct 17 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Better description from Frank Ellermann, Jun 15 2001
STATUS
approved