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). |
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). |
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 |
Mathematica | Table[With[{r = First /@ FactorInteger@ n}, Select[Range@ n, |
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.