

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
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OFFSET

1,2


COMMENTS

Nondisjoint 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 is an infinite number of n verifing: 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 in the sequence because 7 is prime, so it has only one prime divisor.
8 and 9 are in the sequence because neither of them has two distinct prime divisors.
30 is in the sequence because it is the product of three primes.
On the other hand, 35 is not in the sequence 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


CROSSREFS

Cf. A001358, A100959, A006881, A007774, A001221, A001222, A007304, A000977, A001248, A000961.
Sequence in context: A196736 A284946 A285901 * A212165 A319161 A130091
Adjacent sequences: A246713 A246714 A246715 * A246717 A246718 A246719


KEYWORD

nonn,easy


AUTHOR

Giuseppe Coppoletta, Nov 01 2014


STATUS

approved



