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

1,5

Comments

F(n) = {d|n, d>=sqrt(n)}. S(d) = {x|x is a prime number<d, not(x | d)} then a(n) = Sum_{F(n)} card(S(d)). - Devansh 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

Links

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

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

Example

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.

Prog

(PARI) a(n) = #select(x->((x<n) && (n%x)), setbinop((x, y)->(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

Crossrefs

Cf. A000720, A001221.

Sequence in context: A029121 A161205 A340619 * A228942 A164531 A138161

Adjacent sequences: A334797 A334798 A334799 * A334801 A334802 A334803

Keyword

nonn

Author

Devansh Singh, May 12 2020

Extensions

More terms from Michel Marcus, May 13 2020

Status

approved