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

%I A092695 #32 Sep 11 2023 09:11:03
%S A092695 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,
%T A092695 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,
%U A092695 14,14,15,15,15,15,15,15,16,16,16,16,17,17,18,18,18,18,18,18,19,19,19,19
%N A092695 Number of positive integers less than or equal to n which are not divisible by the primes 2,3,5,7.
%C A092695 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
%D A092695 John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 62.
%H A092695 Reinhard Zumkeller, <a href="/A092695/b092695.txt">Table of n, a(n) for n = 0..10000</a>
%H A092695 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%F A092695 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.
%F A092695 a(n + 210) = a(n) + 48.
%F A092695 a(n) = -a(-1 - n) for n < 0.
%F A092695 a(n) ~ (8/35)*n. - _Amiram Eldar_, Dec 06 2020
%e A092695 x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + 2*x^11 + ...
%t A092695 Accumulate @ Table[Boole @ CoprimeQ[n, 210], {n, 0, 100}] (* _Amiram Eldar_, Dec 06 2020 *)
%o A092695 (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}
%o A092695 (PARI) {a(n) = if( n<0, -a(-1 - n), sum( k=0, n, 1==gcd( k, 210)))}
%o A092695 (Haskell)
%o A092695 a092695 n = a092695_list !! n
%o A092695 a092695_list = scanl (+) 0 $
%o A092695                map (fromEnum . (> 7)) (8 : tail a020639_list)
%o A092695 -- _Reinhard Zumkeller_, Mar 26 2012
%Y A092695 Cf. A020639, A008364.
%K A092695 nonn
%O A092695 0,12
%A A092695 _Michael Somos_, Mar 04 2004