

A143145


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).


3



1, 2, 4, 6, 9, 12, 15, 16, 20, 25, 30, 35, 36, 42, 49, 56, 63, 64, 70, 72, 77, 80, 81, 88, 90, 99, 100, 110, 121, 132, 143, 144, 156, 169, 182, 195, 196, 208, 210, 221, 224, 225, 238, 240, 255, 256, 272, 289, 306, 323, 324, 342, 361, 380, 399, 400, 418, 420, 437, 440
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OFFSET

1,2


COMMENTS

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).
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.
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


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000


EXAMPLE

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.


MATHEMATICA

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


CROSSREFS

Cf. A143143, A143144.
Sequence in context: A189753 A278450 A030763 * A328212 A352191 A256393
Adjacent sequences: A143142 A143143 A143144 * A143146 A143147 A143148


KEYWORD

nonn


AUTHOR

Leroy Quet, Jul 27 2008


EXTENSIONS

More terms from Robert G. Wilson v, Aug 31 2008


STATUS

approved



