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A246716
Positive numbers that are not the product of (exactly) two distinct primes.
3
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 100
OFFSET
1,2
COMMENTS
Non-disjoint union of A100959 and A000961. Disjoint union of A100959 and A001248.
Complement of A006881, then inheriting the "opposite" of the properties of A006881.
a(n+1) - a(n) <= 4 (gap upper bound) - (that is because among four consecutive integers there is always a multiple of 4, then there is a number which is not the product of two distinct primes). E.g., a(26)-a(25) = a(62)-a(61) = 4. Is it true that for any k <= 4 there are infinitely many numbers n such that a(n+1) - a(n) = k?
If r = A006881(n+1) - A006881(n) - 1 > 1, it indicates that there are r terms of (a(j)) starting with j = A006881(n) - n + 1 which are consecutive integers. E.g., A006881(8) - A006881(7) - 1 = 6, so there are 6 consecutive terms in (a(j)), starting with j = A006881(7) - 7 + 1 = 20.
LINKS
EXAMPLE
7 is a term because 7 is prime, so it has only one prime divisor.
8 and 9 are terms because neither of them has two distinct prime divisors.
30 is a term because it is the product of three primes.
But 35 is not a term because it is the product of two distinct primes.
MAPLE
filter:= n -> map(t -> t[2], ifactors(n)[2]) <> [1, 1]:
select(filter, [$1..1000]); # Robert Israel, Nov 02 2014
MATHEMATICA
Select[Range[125], Not[PrimeOmega[#] == PrimeNu[#] == 2] &] (* Alonso del Arte, Nov 03 2014 *)
PROG
(PARI) isok(n) = (omega(n)!=2) || (bigomega(n) != 2); \\ Michel Marcus, Nov 01 2014
(Magma) [n: n in [1..100] | #PrimeDivisors(n) ne 2 or &*[t[2]: t in Factorization(n)] ne 1]; // Bruno Berselli, Nov 12 2014
(Sage)
def A246716_list(n) :
R = []
for i in (1..n) :
d = prime_divisors(i)
if len(d) != 2 or d[0]*d[1] != i : R.append(i)
return R
A246716_list(100)
(Sage) [n for n in (1..100) if sloane.A001221(n)!=2 or sloane.A001222(n)!=2] # Giuseppe Coppoletta, Jan 19 2015
(Python)
from math import isqrt
from sympy import primepi, primerange
def A246716(n):
def f(x): return int(n-(t:=primepi(s:=isqrt(x)))-(t*(t-1)>>1)+sum(primepi(x//k) for k in primerange(1, s+1)))
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
return bisection(f) # Chai Wah Wu, Aug 30 2024
KEYWORD
nonn,easy
AUTHOR
Giuseppe Coppoletta, Nov 01 2014
STATUS
approved