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A283427
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a(n) is the number of consecutive smallest prime totatives of primorial A002110(n).
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1
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0, 1, 7, 26, 34, 55, 65, 91, 137, 152, 208, 251, 270, 315, 394, 471, 502, 591, 656, 685, 790, 864, 977, 1139, 1227, 1268, 1354, 1395, 1494, 1847, 1945, 2109, 2157, 2455, 2512, 2693, 2878, 3005, 3202, 3396, 3471, 3826, 3902, 4045, 4119, 4581, 5059, 5226, 5307
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OFFSET
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1,3
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COMMENTS
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Let p_n# = A002110(n) be the n-th primorial, and let t be a totative of p_n#, i.e., gcd(t, p_n#) = 1. Let q be the smallest prime totative of p_n#. We know q must be p_(n+1) by the definition of "primorial" as the product of the smallest n primes. This is the starting point of the range of primes we are considering. The ending point is the smallest composite totative, which is a square semiprime. This semiprime in fact must be q^2, since q is the smallest prime totative of p_n#. Stated in terms of prime n, the range we are considering are primes p_(n+1) <= t <= prevprime((p_(n+1))^2). For the smallest primorials, q^2 > p_n# with n <= 3. Thus a(n) < A054272(n) for n <= 3.
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LINKS
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FORMULA
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a(n) = pi(min(prime(n+1)^2, Product_{k=1..n} ( prime(k) ) )) - n.
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EXAMPLE
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a(2) = pi(min(prime(3)^2, p_2#)) - 2 = pi(min(25,6)) - 2 = 3 - 2 = 1.
a(4) = pi(min(prime(5)^2, p_4#)) - 4 = pi(min(121,210)) - 4 = 30 - 4 = 26.
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MATHEMATICA
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Table[PrimePi[Min[Prime[n + 1]^2, Product[Prime@ i, {i, n}]]] - n, {n, 49}] (* Michael De Vlieger, May 16 2017 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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