A272618 - OEIS

%I #14 May 23 2016 13:11:36

%S 0,0,0,0,0,4,0,0,0,4,8,0,8,9,0,4,8,9,0,0,4,8,12,16,0,8,16,9,4,8,16,0,

%T 9,16,18,0,4,8,16,0,8,16,0,4,8,9,12,16,18,20,24,25,27,0,0,9,27,4,8,16,

%U 32,25,8,16,24,27,32,0,4,8,16,32,9,27,16,25,32,0,4,8,9,12,16,18,24,27,28,32

%N Irregular array read by rows: n-th row contains (in ascending order) the nondivisors 1 <= k < n such that all the prime divisors p of k also divide n.

%C The k are the "semidivisors" or nondivisor regular numbers of n as counted by A243822(n).

%C All nonzero terms k are composite and pertain to composite rows n. This is because prime k must either divide or be coprime to n, and k = 1 is both a divisor of and coprime to n.

%C Row n for prime p contains zero, since numbers 1 <= k < p must either divide or be coprime to prime p.

%C Row n for prime powers p^e contains zero, since there is only one prime divisor p of p^e and every power 1 <= m <= e of p divides p^e.

%C Row n = 4 is a special case of composite n that contains zero. This is because 4 is the smallest composite number; there are no composites k < n.

%C Thus rows n for composite n > 4 contain at least 1 nonzero value.

%C In base n, 1/a(n) has a terminating expansion with at least 2 places.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, pp. 144-145, Theorem 136.

%H Michael De Vlieger, <a href="/A272618/b272618.txt">Table of n, a(n) for n = 1..10814</a> (rows 1 to 1000, flattened).

%H M. De Vlieger, <a href="http://dx.doi.org/10.1145/2077808.2077809">Exploring Number Bases as Tools</a>, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.

%H M. De Vlieger, <a href="http://www.vincico.com/proof/neutral.html">Neutral Numbers</a>.

%H M. De Vlieger, <a href="http://www.vincico.com/seq/a272618.html">Sequence page</a>.

%e For n = 12, the numbers 1 <= k < n such that the prime divisors p of k also divide n are {2, 3, 4, 6, 8, 9}; {2, 3, 4, 6} divide n = 12, thus row n = 12 is {8, 9}.

%e n: k

%e 1: 0

%e 2: 0

%e 3: 0

%e 4: 0

%e 5: 0

%e 6: 4

%e 7: 0

%e 8: 0

%e 9: 0

%e 10: 4 8

%e 11: 0

%e 12: 8 9

%e 13: 0

%e 14: 4 8

%e 15: 9

%e 16: 0

%e 17: 0

%e 18: 4 8 12 16

%e 19: 0

%e 20: 8 16

%t Table[With[{r = First /@ FactorInteger@ n}, Select[Range@ n,

%t And[SubsetQ[r, Map[First, FactorInteger@ #]], ! Divisible[n, #]] &]], {n, 30}] /. {} -> 0 // Flatten (* _Michael De Vlieger_, May 03 2016 *)

%Y Union of A027750 and nonzero terms of a(n) = A162306, thus A000005(n) + A243822(n) = A010846(n).

%Y The union of nonzero terms of a(n) and A272619 = A133995, thus A243822(n) + A243823(n) = A045763(n).

%K nonn,tabf

%O 1,6

%A _Michael De Vlieger_, May 03 2016