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A092695
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Number of positive integers less than or equal to n which are not divisible by the primes 2,3,5,7.
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8
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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
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0,12
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COMMENTS
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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
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REFERENCES
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John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 62.
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LINKS
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FORMULA
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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.
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EXAMPLE
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x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + 2*x^11 + ...
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MATHEMATICA
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Accumulate @ Table[Boole @ CoprimeQ[n, 210], {n, 0, 100}] (* Amiram Eldar, Dec 06 2020 *)
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PROG
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(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
a092695_list = scanl (+) 0 $
map (fromEnum . (> 7)) (8 : tail a020639_list)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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