A092695 Number of positive integers less than or equal to n which are not divisible by the primes 2,3,5,7.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19

Offset: 0

Author

Michael Somos, Mar 04 2004

Keywords

• nonn

Comments

This sequence is a special case of the following: Take different primes p_1, p_2,...,p_k. For a nonempty subset I of {1,2,...,k} denote by |I| the number of its elements. For a positive integer n denote A(n,I) = floor(n/Product_{i in I} p_i). Then the number of positive integers m <= n such that m is divisible by none of p_1,p_2,...,p_k is equal to n + Sum_{} (-1)^(|I|)*A(n,I), where I runs over all nonempty subsets of {1,2,...,k}. - Milan Janjic, Apr 23 2007

Examples

			x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + 2*x^11 + ...
		

References

• John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 62.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for two-way infinite sequences

Crossrefs

Cf. A020639, A008364.

Programs

Haskell

a092695 n = a092695_list !! n
a092695_list = scanl (+) 0 $
               map (fromEnum . (> 7)) (8 : tail a020639_list)
-- Reinhard Zumkeller, Mar 26 2012

Mathematica

Accumulate @ Table[Boole @ CoprimeQ[n, 210], {n, 0, 100}] (* Amiram Eldar, Dec 06 2020 *)

PARI

{a(n) = n - n\2 - n\3 - n\5 - n\7 + n\6 + n\10 + n\14 + n\15 + n\21 - n\30 + n\35 - n\42 - n\70 - n\105 + n\210}

PARI

{a(n) = if( n<0, -a(-1 - n), sum( k=0, n, 1==gcd( k, 210)))}

Formula

G.f.: (x * P172 * P36) / (e(1) * e(210)) where e(n) = 1 - x^n, P36 = e(16) * e(20) * e(24) / (e(6) * e(8) * e(10)) is a polynomial of degree 36 and P172 is a polynomial of degree 172.

a(n + 210) = a(n) + 48.

a(n) = -a(-1 - n) for n < 0.

a(n) ~ (8/35)*n. - Amiram Eldar, Dec 06 2020