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%I A272619 #13 May 23 2016 13:11:41
%S A272619 0,0,0,0,0,0,0,6,6,6,0,10,0,6,10,12,6,10,12,6,10,12,14,0,10,14,15,0,6,
%T A272619 12,14,15,18,6,12,14,15,18,6,10,12,14,18,20,0,10,14,15,20,21,22,10,15,
%U A272619 20,6,10,12,14,18,20,22,24,6,12,15,18,21,24,6,10,12,18,20,21,22,24,26,0,14,21,22
%N A272619 Irregular array read by rows: n-th row contains (in ascending order) the numbers 1 <= k < n such that at least one prime divisor p of k also divides n and at least one prime divisor q of k is coprime to n.
%C A272619 The k are the "semitotatives" of n as counted by A243823(n).
%C A272619 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. Further, the terms k must have at least two distinct prime divisors p and q.
%C A272619 Row n for prime p contains zero, since numbers 1 <= k < p must either divide or be coprime to prime p.
%C A272619 Row n for prime powers p^e contains all the numbers k in the corresponding row of A133995. There is only one prime divisor p of p^e and every power 1 <= m <= e of p divides p^e, thus none of the terms of the corresponding row of A133995 are in A272618(n).
%C A272619 Rows n = 4 and 6 are special cases of composite n that contains zero. 4 is the smallest composite number; there are no composites k < n. 6 has the prime divisors 2 and 3, thus 5 is the smallest prime coprime to 6; the product of the minimum prime divisor and minimum prime coprime to 6 is 10, which exceeds 6 and falls outside the considered range. The situation is not so for composite n > 6. Thus rows n for composite n > 6 contain at least 1 nonzero value.
%C A272619 The smallest k of row n = A096014(n) < n, i.e., those values of A096014(n) pertaining to composite n > 6, a product of the smallest prime divisor p of n and the smallest prime q coprime to n. The smallest k of n are even squarefree semiprimes since 2 either divides n or is coprime to n and k is by definition a number with at least two distinct primes. The smallest k = 2p for p^2 sets record values for A096014(n) when we ignore values pertaining to prime n, n = 4, and n = 6.
%C A272619 In base n, 1/a(n) has a mixed recurrent expansion.
%D A272619 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, pp. 144-5, Theorem 136.
%H A272619 Michael De Vlieger, <a href="/A272619/b272619.txt">Table of n, a(n) for n = 1..10447</a> (rows 1 to 256, flattened).
%H A272619 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 A272619 M. De Vlieger, <a href="http://www.vincico.com/proof/neutral.html">Neutral Numbers</a>.
%H A272619 M. De Vlieger, <a href="http://www.vincico.com/proof/a272619.html">Sequence page</a>.
%e A272619 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 A272619 n:   k
%e A272619 1:   0
%e A272619 2:   0
%e A272619 3:   0
%e A272619 4:   0
%e A272619 5:   0
%e A272619 6:   0
%e A272619 7:   0
%e A272619 8:   6
%e A272619 9:   6
%e A272619 10:  6
%e A272619 11:  0
%e A272619 12: 10
%e A272619 13:  0
%e A272619 14:  6 10 12
%e A272619 15:  6 10 12
%e A272619 16:  6 10 12 14
%e A272619 17:  0
%e A272619 18: 10 14 15
%e A272619 19:  0
%e A272619 20:  6 12 14 15 18
%t A272619 Table[With[{r = First /@ FactorInteger@ n}, Select[Range@ n, Function[m, And[! SubsetQ[r, First /@ FactorInteger@ m], 1 < GCD[m, n] < n]]]], {n, 30}] /. {} -> {0} // Flatten (* _Michael De Vlieger_, May 03 2016 *)
%Y A272619 The union of nonzero terms of a(n) and A272618 = A133995, thus A243822(n) + A243823(n) = A045763(n).
%K A272619 nonn,tabf
%O A272619 1,8
%A A272619 _Michael De Vlieger_, May 03 2016

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