A158974 -id:A158974 - cp's OEIS frontend

A158976 a(n) = sum of numbers k <= n such that not all proper divisors of k are divisors of n.

0, 0, 0, 0, 4, 0, 10, 6, 18, 23, 37, 10, 49, 45, 54, 66, 94, 75, 112, 90, 123, 149, 175, 120, 199, 220, 241, 251, 305, 236, 335, 307, 358, 396, 409, 385, 505, 501, 534, 499, 622, 568, 664, 630, 632, 749, 799, 688, 847, 857, 937, 959, 1049, 985, 1078, 1039, 1205

Offset: 1

Jaroslav Krizek, Apr 01 2009

nonn

For primes p, a(p) = A000217(p) - A158662(p) = A000217(p) - A014284(A036234(p)).

			For n = 7 we have the following proper divisors for k <= n: {1}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}. Only 4 and 6 have proper divisors that are not divisors of 7, viz. 2 and 2, 3. Hence a(7) = 4 + 6 = 10.

Cf. A000040, A000217, A158662, A014284, A036234, A158974.

[ IsEmpty(S) select 0 else &+S where S is [ k: k in [1..n] | exists(t){ d: d in Divisors(k) | d ne k and d notin Divisors(n) } ]: n in [1..57] ];

Edited and extended by Klaus Brockhaus, Apr 06 2009

A159072 Count of numbers k in the range 1<=k<= n such that set of proper divisors of k is not a subset of the set of the proper divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 2, 7, 6, 7, 7, 10, 7, 11, 8, 11, 12, 14, 8, 15, 15, 16, 15, 19, 13, 20, 17, 20, 21, 22, 17, 25, 24, 25, 21, 28, 23, 29, 26, 26, 30, 32, 24, 33, 31, 34, 33, 37, 32, 37, 33, 39, 40, 42, 32, 43, 42, 40, 41, 45, 42, 48, 45, 48, 44, 51, 41, 52, 51, 50, 51, 54

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.

We use the (nonstandard) terminology that the empty set (the proper divisors of 1) is not a subset of another set.

			a(8) = 2 counts k=6 with divisors set {1, 2, 3} (not subset of the divisors {1, 2, 4} of n = 8), and k=1 without proper divisors.

Cf.: A158974, A000040, A036234, A000720.

a(n)+A159070(n) = n. - R. J. Mathar, Apr 06 2009

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

cp's OEIS Frontend

A158976 a(n) = sum of numbers k <= n such that not all proper divisors of k are divisors of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Extensions