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 A272618 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. 21
 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 (list; graph; refs; listen; history; text; internal format)
 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 with 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. 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: 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 03 2016 *) CROSSREFS Union of A027750 and nonzero terms of a(n) = A162306, thus A000005(n) + A243822(n) = A010846(n). The union of nonzero terms of a(n) and A272619 = A133995, thus A243822(n) + A243823(n) = A045763(n). Sequence in context: A173532 A270029 A028598 * A265397 A071327 A071326 Adjacent sequences:  A272615 A272616 A272617 * A272619 A272620 A272621 KEYWORD nonn,tabf AUTHOR Michael De Vlieger, May 03 2016 STATUS approved

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Last modified October 15 21:17 EDT 2019. Contains 328038 sequences. (Running on oeis4.)