OEIS A272619

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 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.

Data

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, 12, 14, 15, 18, 6, 12, 14, 15, 18, 6, 10, 12, 14, 18, 20, 0, 10, 14, 15, 20, 21, 22, 10, 15, 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

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.

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 3 2016 *)

Crossrefs

The union of nonzero terms of this sequence and A272618 = A133995, thus A243822(n) + A243823(n) = A045763(n).

Keyword

nonn, tabf

Author Michael De Vlieger, May 3 2016

This sequence lists the semitotatives k of n in numerical order. A semitotative is a composite 1 < k < n with at least one prime divisor p that also divides n, and at least one prime divisor q coprime to n. Semitotatives are neutral numbers m with respect to n, thus a composite m that neither divides nor is coprime to n. Thus, prime k cannot be a semitotative, and prime n can have no semitotatives. 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 semitotatives. Also, n = 6 has no semitotatives because the product 10 of the minimum prime divisor p = 2 and the minimum prime totative q = 5 exceeds 6. The k of row n are counted by A243823. 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.