OFFSET
1,1
COMMENTS
Numbers k such that phi(k) + sigma(k) = 2*(k+1). - Benoit Cloitre, Mar 02 2002
Numbers k such that tau(k) = omega(k)^omega(k). - Benoit Cloitre, Sep 10 2002 [This comment is false. If k = 900 then tau(k) = omega(k)^omega(k) = 27 but 900 = (2*3*5)^2 is not the product of two distinct primes. - Peter Luschny, Jul 12 2023]
Could also be called 2-almost primes. - Rick L. Shepherd, May 11 2003
From the Goldston et al. reference's abstract: "lim inf [as n approaches infinity] [(a(n+1) - a(n))] <= 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6." - Jonathan Vos Post, Jun 20 2005
The maximal number of consecutive integers in this sequence is 3 - there cannot be 4 consecutive integers because one of them would be divisible by 4 and therefore would not be product of distinct primes. There are several examples of 3 consecutive integers in this sequence. The first one is 33 = 3 * 11, 34 = 2 * 17, 35 = 5 * 7; (see A039833). - Matias Saucedo (solomatias(AT)yahoo.com.ar), Mar 15 2008
Number of terms less than or equal to 10^k for k >= 0 is A036351(k). - Robert G. Wilson v, Jun 26 2012
Are these the numbers k whose difference between the sum of proper divisors of k and the arithmetic derivative of k is equal to 1? - Omar E. Pol, Dec 19 2012
A237114(n) (smallest semiprime k^prime(n)+1) is a term, for n != 2. - Jonathan Sondow, Feb 06 2014
a(n) are the reduced denominators of p_2/p_1 + p_4/p_3, where p_1 != p_2, p_3 != p_4, p_1 != p_3, and the p's are primes. In other words, (p_2*p_3 + p_1*p_4) never shares a common factor with p_1*p_3. - Richard R. Forberg, Mar 04 2015
Conjecture: The sums of two elements of a(n) forms a set that includes all primes greater than or equal to 29 and all integers greater than or equal to 83 (and many below 83). - Richard R. Forberg, Mar 04 2015
A263990 lists the subsequence of a(n) where a(n+1)=1+a(n). - R. J. Mathar, Aug 13 2019
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zervos, Marie: Sur une classe de nombres composés. Actes du Congrès interbalkanique de mathématiciens 267-268 (1935)
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pimtz and Y. Yildirim, "Small Gaps Between Primes or Almost Primes", arXiv:math/0506067 [math.NT], March 2005.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
R. J. Mathar, Series of reciprocal powers of k-almost primes arXiv:0803.0900, table 6 k=2 shows sum 1/a(n)^s.
Eric Weisstein's World of Mathematics, Semiprime
FORMULA
A056595(a(n)) = 3. - Reinhard Zumkeller, Aug 15 2011
A211110(a(n)) = 3. - Reinhard Zumkeller, Apr 02 2012
Sum_{n >= 1} 1/a(n)^s = (1/2)*(P(s)^2 - P(2*s)), where P is Prime Zeta. - Enrique Pérez Herrero, Jun 24 2012
A050326(a(n)) = 2. - Reinhard Zumkeller, May 03 2013
sopf(a(n)) = a(n) - phi(a(n)) + 1 = sigma(a(n)) - a(n) - 1. - Wesley Ivan Hurt, May 18 2013
d(a(n)) = 4. Omega(a(n)) = 2. omega(a(n)) = 2. mu(a(n)) = 1. - Wesley Ivan Hurt, Jun 28 2013
a(n) ~ n log n/log log n. - Charles R Greathouse IV, Aug 22 2013
A089233(a(n)) = 1. - Reinhard Zumkeller, Sep 04 2013
From Peter Luschny, Jul 12 2023: (Start)
For k > 1: k is term of a <=> A363923(k) = k. (End)
MAPLE
N:= 1001: # to get all terms < N
Primes:= select(isprime, [2, seq(2*k+1, k=1..floor(N/2))]):
{seq(seq(p*q, q=Primes[1..ListTools:-BinaryPlace(Primes, N/p)]), p=Primes)} minus {seq(p^2, p=Primes)};
# Robert Israel, Jul 23 2014
# Alternative, using A001221:
isA006881 := proc(n)
if numtheory[bigomega](n) =2 and A001221(n) = 2 then
true ;
else
false ;
end if;
end proc:
A006881 := proc(n) if n = 1 then 6; else for a from procname(n-1)+1 do if isA006881(a) then return a; end if; end do: end if;
end proc: # R. J. Mathar, May 02 2010
# Alternative:
with(NumberTheory): isA006881 := n -> is(NumberOfPrimeFactors(n, 'distinct') = 2 and NumberOfPrimeFactors(n) = 2):
select(isA006881, [seq(1..205)]); # Peter Luschny, Jul 12 2023
MATHEMATICA
mx = 205; Sort@ Flatten@ Table[ Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[ mx/Prime[n]]}] (* Robert G. Wilson v, Dec 28 2005, modified Jul 23 2014 *)
sqFrSemiPrimeQ[n_] := Last@# & /@ FactorInteger@ n == {1, 1}; Select[Range[210], sqFrSemiPrimeQ] (* Robert G. Wilson v, Feb 07 2012 *)
With[{upto=250}, Select[Sort[Times@@@Subsets[Prime[Range[upto/2]], {2}]], #<=upto&]] (* Harvey P. Dale, Apr 30 2018 *)
PROG
(PARI) for(n=1, 214, if(bigomega(n)==2&&omega(n)==2, print1(n, ", ")))
(PARI) for(n=1, 214, if(bigomega(n)==2&&issquarefree(n), print1(n, ", ")))
(PARI) list(lim)=my(v=List()); forprime(p=2, sqrt(lim), forprime(q=p+1, lim\p, listput(v, p*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
(Haskell)
a006881 n = a006881_list !! (n-1)
a006881_list = filter chi [1..] where
chi n = p /= q && a010051 q == 1 where
p = a020639 n
q = n `div` p
-- Reinhard Zumkeller, Aug 07 2011
(Sage)
def A006881_list(n) :
R = []
for i in (6..n) :
d = prime_divisors(i)
if len(d) == 2 :
if d[0]*d[1] == i :
R.append(i)
return R
A006881_list(205) # Peter Luschny, Feb 07 2012
(Magma) [n: n in [1..210] | EulerPhi(n) + DivisorSigma(1, n) eq 2*(n+1)]; // Vincenzo Librandi, Sep 17 2015
(Python)
from sympy import factorint
def ok(n): f=factorint(n); return len(f) == 2 and sum(f[p] for p in f) == 2
print(list(filter(ok, range(1, 206)))) # Michael S. Branicky, Jun 10 2021
(Python)
from math import isqrt
from sympy import primepi, primerange
def A006881(n):
def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
return m # Chai Wah Wu, Aug 15 2024
CROSSREFS
Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A030229, A051709, A001221 (omega(n)), A001222 (bigomega(n)), A001358 (semiprimes), A005117 (squarefree), A007304 (squarefree 3-almost primes), A213952, A039833, A016105 (subsequences), A237114 (subsequence, n != 2).
Subsequence of A007422.
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Name expanded (based on a comment of Rick L. Shepherd) by Charles R Greathouse IV, Sep 16 2015
STATUS
approved