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A328570
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Index of the least significant zero digit in the primorial base expansion of n, when the rightmost digit is in the position 1.
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10
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1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5
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OFFSET
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0,2
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COMMENTS
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Index of the least prime not dividing A276086(n), where A276086 converts the primorial base expansion of n into its prime product form.
Starting from x = n, repeatedly divide x by prime(1) (discarding the remainder), and set x to the integer quotient floor(x/prime(1)), then divide x with prime(2) (again discarding the remainder, and setting x to the integer quotient), etc., stopping as soon one of the primes is a divisor of the previous integer quotient (leaving zero remainder). a(n) is then the index of that prime, equal to 1 + the number of iterations done.
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LINKS
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FORMULA
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EXAMPLE
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For n = 2, we divide it with A000040(1) = 2, and it leaves zero remainder, so we have finished on the first round (needing no actual iterations), and thus a(2) = 1. Note that 2 in primorial base (A049345) is written as "10", and indeed the first zero from the right occurs at the position 1.
For n = 5, we first divide 5 with prime(1) = 2, and discarding the remainder, we are left with floor(5/2) = 2. Then we divide that 2 with prime(2) = 3, leaving floor(2/3) = 0 and remainder 2. And finally we divide 0 with prime(3) = 5, and that doesn't leave any remainder, thus we are finished on the third round, and a(5) = 3. Note that 5 in primorial base is written as "21", and allowing here a leading zero, written as "021", we see that it is in this case the least significant zero occurring at position 3 from the right.
For n = 43, we first divide it with prime(1) = 2, leaving a remainder 1 and integer quotient 21. Then we divide 21 with prime(2) = 3, which doesn't leave any remainder, thus we are finished on the second round, and a(43) = 2. Note that 43 is written as "1201" in primorial base, with the least significant zero occurring in the position 2.
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PROG
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(PARI) A328570(n) = { my(i=1, p=2); while(n && (n%p), i++; n = n\p; p = nextprime(1+p)); (i); };
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CROSSREFS
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Cf. A000720, A002110, A049345, A053669, A055396, A143293, A257993, A276086, A276087, A276088, A326810, A328475, A328476, A328569, A328578, A328580.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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