A143145 - OEIS

%I #13 Mar 08 2024 09:01:41

%S 1,2,4,6,9,12,15,16,20,25,30,35,36,42,49,56,63,64,70,72,77,80,81,88,

%T 90,99,100,110,121,132,143,144,156,169,182,195,196,208,210,221,224,

%U 225,238,240,255,256,272,289,306,323,324,342,361,380,399,400,418,420,437,440

%N A positive integer n is included if there are no primes p where j < p < k, where j is the largest divisor of n that is <= sqrt(n) and k = the smallest divisor of n that is >= sqrt(n).

%C All squares s are included in this sequence, since there are no integers at all between j and k because j = k, where j = the largest divisor of s that is <= sqrt(s) and k the smallest divisor of s that is >= sqrt(s).

%C Also, all integers of the form m = j*(j+1) are included in the sequence, because the two middle divisors are j and j+1 and there are no integers between these divisors, obviously.

%C The number of terms less than or equal to 10^n, n=0..., is 1, 5, 27, 100, 388, 1536, ..., . - _Robert G. Wilson v_, Aug 31 2008

%H Amiram Eldar, <a href="/A143145/b143145.txt">Table of n, a(n) for n = 1..10000</a>

%e The divisors of 35 are 1,5,7,35. The two middle divisors are 5 and 7. Between 5 and 7 (and not including 5 and 7) there are no primes (since the only integer between these divisors, 6, is composite). So 35 is included in the sequence.

%t fQ[n_] := If[ IntegerQ@ Sqrt@ n, True, Block[ {d = Divisors@ n}, len = Length@ d; lst = Take[ PrimeQ@ Range[ d[[len/2]], d[[len/2 + 1]]], {2, -2}]; lst == {} || Union[ lst][[ -1]] != True]]; Select[ Range@ 459, fQ@# &] (* _Robert G. Wilson v_, Aug 31 2008 *)

%Y Cf. A143143, A143144.

%K nonn

%O 1,2

%A _Leroy Quet_, Jul 27 2008

%E More terms from _Robert G. Wilson v_, Aug 31 2008