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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.
2
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
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
Sequence in context: A029121 A161205 A340619 * A228942 A164531 A138161
KEYWORD
nonn
AUTHOR
Devansh Singh, May 12 2020
EXTENSIONS
More terms from Michel Marcus, May 13 2020
STATUS
approved