A158973 -id:A158973 - cp's OEIS frontend

A158974 a(n) is the number of numbers k <= n such that not all proper divisors of k are divisors of n.

0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 1, 6, 5, 6, 6, 9, 6, 10, 7, 10, 11, 13, 7, 14, 14, 15, 14, 18, 12, 19, 16, 19, 20, 21, 16, 24, 23, 24, 20, 27, 22, 28, 25, 25, 29, 31, 23, 32, 30, 33, 32, 36, 31, 36, 32, 38, 39, 41, 31, 42, 41, 39, 40, 44, 41, 47, 44, 47, 43, 50, 40, 51, 50, 49, 50

Offset: 1

Jaroslav Krizek, Apr 01 2009

nonn

			For n = 8 we have the following proper divisors for k <= n: {1}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}, {1, 2, 4}. Only k = 6 has a proper divisor that is not a divisor of 8, viz. 3. Hence a(8) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A036234, A000720, A158973.

[ #[ k: k in [1..n] | exists(t){ d: d in Divisors(k) | d ne k and d notin Divisors(n) } ]: n in [1..76] ];

f:= proc(n) local d;
   d:= numtheory:-divisors(n);
   nops(remove(t -> (numtheory:-divisors(t) minus {t}) subset d, [$4..n-1]))
end proc:
map(f, [$1..100]); # Robert Israel, Mar 30 2020

a[n_] := Select[Most[Divisors[#]]& /@ Range[n], AnyTrue[#, !Divisible[n, #]&]&] // Length;
Array[a, 100] (* Jean-François Alcover, Jul 17 2020 *)

a(n) = my(dn = divisors(n)); sum(k=1, n, my(dk=setminus(divisors(k), Set(k))); #setintersect(dk, dn) != #dk); \\ Michel Marcus, Aug 27 2020

For primes p, a(p) = p - A036234(p) = p - A000720(p) - 1.

Edited and extended by Klaus Brockhaus, Apr 06 2009

A159070 Count of numbers k in the range 1 < k <= n such that set of proper divisors of k is a subset of the set of proper divisors of n.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 6, 5, 6, 5, 10, 6, 8, 8, 9, 7, 11, 8, 12, 10, 10, 9, 16, 10, 11, 11, 13, 10, 17, 11, 15, 13, 13, 13, 19, 12, 14, 14, 19, 13, 19, 14, 18, 19, 16, 15, 24, 16, 19, 17, 19, 16, 22, 18, 23, 18, 18, 17, 28, 18, 20, 23, 23, 20, 24, 19, 23, 21, 26, 20, 31, 21, 23, 25, 25

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

Here proper divisors include 1 but not the argument (k or n, respectively) in the divisor set, as defined in A032741.

			a(8) = 6 admits the following 6 k: 2 {1}, 3 {1}, 4 {1, 2}, 5 {1}, 7 {1}, 8 {1, 2, 4} with subsets of the proper divisors {1, 2, 4} for n = 8.

Cf. A158973, A000040, A036234, A000720.

a(n) = A158973(n) - 1.

If p = prime, element of A000040, a(p) = A158973(p) - 1 = A036234(p) - 1 = A000720(p).

Edited and extended by R. J. Mathar, Apr 06 2009

cp's OEIS Frontend

A158974 a(n) is the number of numbers k <= n such that not all proper divisors of k are divisors of n.

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