OFFSET
1,8
COMMENTS
The k are the "semitotatives" of n as counted by A243823(n).
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.
Row n for prime p contains zero, since numbers 1 <= k < p must either divide or be coprime to prime p.
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).
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.
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.
In base n, 1/a(n) has a mixed recurrent expansion.
REFERENCES
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.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10447 (rows 1 to 256, flattened).
M. De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
M. De Vlieger, Neutral Numbers.
M. De Vlieger, Sequence page.
EXAMPLE
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}.
n: k
1: 0
2: 0
3: 0
4: 0
5: 0
6: 0
7: 0
8: 6
9: 6
10: 6
11: 0
12: 10
13: 0
14: 6 10 12
15: 6 10 12
16: 6 10 12 14
17: 0
18: 10 14 15
19: 0
20: 6 12 14 15 18
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Michael De Vlieger, May 03 2016
STATUS
approved