OEIS A272618

by Michael Thomas De Vlieger, updated 23 May 2016, St. Louis, Missouri. First published 3 May 2016.

Name

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.
Data

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, 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, 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
Offset

1, 6
Comments

The k are the "semidivisors" or nondivisor regular numbers of n as counted by A243822(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.
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 zero, since there is only one prime divisor p of p^e and every power 1 <= m <= e of p divides p^e.
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.
Thus rows n for composite n > 4 contain at least 1 nonzero value.
In base n, 1/a(n) has a terminating expansion in at least 2 places.
References

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.
Links

Michael De Vlieger, Table of n, a(n) for n = 1..10814 (rows 1 to 1000, 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.
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:  4
7:  0
8:  0
9:  0
10: 4  8
11: 0
12: 8  9
13: 0
14: 4  8
15: 9
16: 0
17: 0
18: 4  8 12 16
19: 0
20: 8 16
Mathematica

Table[With[{r = First /@ FactorInteger@ n}, Select[Range@ n,
And[SubsetQ[r, Map[First, FactorInteger@ #]], ! Divisible[n, #]] &]], {n, 30}] /. {} -> 0 // Flatten (* Michael De Vlieger, May 3 2016 *)
Crossrefs

Union of A027750 and nonzero terms of this sequence = A162306(n), thus A000005(n) + A243822(n) = A010846(n).
The union of nonzero terms of this sequence and A272619 = A133995, thus A243822(n) + A243823(n) = A045763(n).
Keyword

nonn, tabf, easy
Author Michael De Vlieger, May 3 2016

This sequence lists the semidivisors k of n in numerical order. A semidivisor is a number 1 < k < n whose prime divisors p also divide n. Semidivisors are nondivisor regular and neutral numbers. A regular number m with respect to n is one that has all prime divisors p that also divide n; all divisors d of n are also regular to n. A neutral number m with respect to n is a composite m that neither divides nor is coprime to n. Since semidivisors are neutral, they must be composite. Thus, prime k cannot be a semidivisor, and prime n can have no semidivisors. Prime powers p^e with e > 1 only have semitotatives but no semidivisors, since prime powers have a single prime divisor p, and all powers 1 < m <= e divide p^e. Since 4 is the smallest composite, n = 4 cannot have semidivisors. The k of row n are counted by A243822. See the entry above for more links to the OEIS pertaining to the identities and related sequences mentioned in this paragraph. See Neutral Numbers for more information regarding neutral and regular numbers with respect to n.

Sequence published by N. J. A. Sloane at OEIS 21 May 2016.