A246716 Positive numbers that are not the product of (exactly) two distinct primes.

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

Giuseppe Coppoletta, Table of n, a(n) for n = 1..10000

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

Crossrefs

Cf. A001358, A100959, A006881, A007774, A001221, A001222, A007304, A000977, A001248, A000961.

Sequence in context: A196736 A284946 A285901 * A374473 A362618 A371782

Adjacent sequences: A246713 A246714 A246715 * A246717 A246718 A246719

Keyword

nonn,easy

Author

Giuseppe Coppoletta, Nov 01 2014

Status

approved