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.

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

Author

Michael De Vlieger, May 03 2016

Keywords

• nonn
• tabf

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.

Examples

			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
		

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.

Crossrefs

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

Programs

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 *)