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A340469
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First constant from family of prime-representing constants h_n (h1 = 1.2148208055...) such that ceiling(h_n) = prime(n).
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3
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1, 2, 1, 4, 8, 2, 0, 8, 0, 5, 5, 2, 4, 3, 3, 3, 7, 4, 6, 9, 4, 5, 1, 3, 1, 2, 3, 4, 2, 2, 3, 7, 7, 0, 9, 5, 4, 2, 5, 9, 1, 5, 0, 2, 6, 0, 2, 1, 2, 2, 7, 2, 3, 9, 1, 5, 8, 0, 4, 1, 4, 6, 1, 9, 2, 9, 3, 8, 1, 3, 9, 9, 0, 9, 7, 9, 7, 6, 3, 2, 6, 0, 6, 0, 4, 0, 5, 9, 0, 6, 0, 3, 3, 3, 5, 5, 6, 2, 6, 5, 6, 3, 9, 0, 8
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OFFSET
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1,2
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COMMENTS
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The family of constants h_n (h1 = 1.2148208055...) for generation of the complete sequence of primes with using of a recursive relation for h_n such that ceiling(h_n) = prime(n). The recursive relation h_n = ceiling(h_{n-1})*(h_{n-1}-ceiling(h_{n-1})+2) generates the complete sequence of prime numbers. Constants h_n are irrational for all n.
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LINKS
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FORMULA
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h1 = Sum_{k>=1} (prime(k)-2)/Product_{i=1..k-1} prime(i).
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EXAMPLE
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h1 = 1.21482080552433374694513123422377095425915026021227...
h2 = 2.42964161104866749389026246844754190851830052042454...
h3 = 4.28892483314600248167078740534262572555490156127363...
etc.
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MATHEMATICA
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N[Sum[(Prime[k]-2)/Product[Prime[n], {n, 1, k-1}], {k, 1, 150}], 50]
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PROG
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(PARI) suminf(k=1, (prime(k)-2)/prod(i=1, k-1, prime(i))) \\ Michel Marcus, Jan 08 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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