A334800 a(n) is the number of values d*p less than n, where d is a divisor of n, p is a prime, and d*p is not a divisor of n.

0, 0, 1, 1, 2, 2, 3, 4, 4, 4, 4, 6, 5, 7, 7, 9, 6, 10, 7, 11, 10, 10, 8, 16, 10, 12, 12, 15, 9, 17, 10, 19, 14, 15, 14, 23, 11, 17, 16, 24, 12, 25, 13, 23, 22, 20, 14, 34, 17, 25, 20, 26, 15, 32, 20, 32, 22, 23, 16, 41, 17, 26, 29, 36, 23, 36, 18, 33, 26, 36

Offset: 1

Author

Devansh Singh, May 12 2020

Keywords

• nonn

Comments

F(n) = {d|n, d>=sqrt(n)}. S(d) = {x|x is a prime numberDevansh Singh, Jun 16 2020

To say "d*p < n and not(d*p | n)" is equivalent to "p < d' and not(p | d')" where d' = n/d also runs over all divisors, whence the formula. - M. F. Hasler, Jun 16 2020

Examples

			a(10)=4 : {3=1*3, 4=2*2, 6=2*3, 7=1*7}, where 1, 2, 5, are excluded because they are divisor of 10; 8 and 9 are excluded because they cannot be written as d*p.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A001221.

Programs

PARI

a(n) = #select(x->((x(x*y), divisors(n), select(x->isprime(x), [1..n]))); \\ Michel Marcus, May 13 2020

PARI

apply( {A334800(n)=sumdiv(n,d,primepi(d)-omega(d))}, [1..99]) \\ M. F. Hasler, Jun 16 2020

Formula

a(n) = A000720(n)-1 = PrimePi(n)-1 when n is a prime number. [Corrected by M. F. Hasler]

a(n) = Sum_{d|n} primepi(d)-omega(d), where omega = A001221. - M. F. Hasler, Jun 16 2020

Extensions

More terms from Michel Marcus, May 13 2020