A062854 First differences of A027424.

1, 2, 3, 3, 5, 4, 7, 5, 6, 6, 11, 6, 13, 8, 9, 8, 17, 9, 19, 10, 12, 12, 23, 10, 16, 14, 15, 13, 29, 12, 31, 15, 18, 18, 20, 13, 37, 20, 21, 16, 41, 17, 43, 20, 21, 24, 47, 17, 31, 22, 27, 23, 53, 22, 31, 22, 30, 30, 59, 19, 61, 32, 28, 26, 36, 26, 67, 30, 36, 26, 71, 23, 73, 38

Offset: 1

Author

Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

Keywords

• nonn

Comments

For prime p, a(p) = p. - Ralf Stephan, Jun 02 2005

a(n) is the number of times n appears in A033677. - Franklin T. Adams-Watters, Nov 18 2005

Conjecture: a(n) > n/log(n) for n > 2. - Thomas Ordowski, Jan 28 2017

a(n) is the number of integers 1<=i<=n such that all divisors of i*n are either <=i or >=n. - Chai Wah Wu, Oct 13 2023

Examples

			a(4)=3 because there are 9 unique products in the 4 X 4 multiplication table (1 2 3 4 6 8 9 12 16), which is 3 more than the 6 unique products in the 3 X 3 multiplication table (1 2 3 4 6 9).
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A027421, A062854, A062855, A062856, A062857, A062859.

Programs

Maple

A062854 := proc(n)
    A027424(n)-A027424(n-1) ;
end proc:
seq(A062854(n),n=1..40) ; # R. J. Mathar, Oct 02 2020

Mathematica

Prepend[Differences@ #, First@ #] &@ Module[{ u = {}}, Table[Length[u = Union[u, n Range@ n]], {n, 100}]] (* Michael De Vlieger, Jan 30 2017 *)

PARI

b(n) = #setbinop((x, y)->x*y, vector(n, i, i); );
a(n) = b(n) - b(n-1); \\ Michel Marcus, Jan 28 2017

Python

from itertools import takewhile
from sympy import divisors
def A062854(n): return sum(1 for i in range(1,n+1) if all(d<=i for d in takewhile(lambda d:dChai Wah Wu, Oct 13 2023

Extensions

More terms from Ralf Stephan, Jun 02 2005