A092695 - OEIS

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

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OFFSET

0,12

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

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

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

EXAMPLE

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

MATHEMATICA

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

PROG

(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)))}

(Haskell)

a092695 n = a092695_list !! n