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A051451
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a(n) = lcm{ 1,2,...,x } where x is the n-th prime power (A000961).
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61
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1, 2, 6, 12, 60, 420, 840, 2520, 27720, 360360, 720720, 12252240, 232792560, 5354228880, 26771144400, 80313433200, 2329089562800, 72201776446800, 144403552893600, 5342931457063200, 219060189739591200
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OFFSET
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1,2
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COMMENTS
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Distinct values of A003418, i.e., A051451 = Union[A003418].
This may be the "smallest" product-based numbering system that has a unique finite representation for every rational number. In this base 1/2 = .1 (1*1/2), 1/3 = .02 (0*1/2 + 2*1/6), 1/5 = .0102 (0*1/2 + 1*1/6 + 0*1/12 + 2*1/60). - Russell Easterly, Oct 03 2001
Partial products of A025473, prime roots of the prime powers.
Conjecture: For every n > 2, there exists a twin prime pair [p, p+2] with p < a(n), such that [a(n)+p, a(n)+p+2] is also a twin prime pair. Example: For n=6 we can take p=11, because for a(6) = 420 is [420+11, 420+13] = [431, 433] also a twin prime pair. This has been verified for 2 < n <= 200. - Mike Winkler, Sep 12 2013, May 09 2014
The prime powers give all values, and do so uniquely. (Other positive integers give repeated values.) - Daniel Forgues, Apr 28 2014
"LCM numeral system": a(n+1) is place value for index n, n >= 0; a(-n+1) is (place value)^(-1) for index n, n < 0. - Daniel Forgues, May 03 2014
Repetitions removed from slowest growing integer series A003418 with integers > 0 converging to 0 in the ring Z^ of profinite integers. Both A003418 and the present sequence may be used as a replacement for the usual "factorial system" for coding profinite integers. - Herbert Eberle, May 01 2016
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LINKS
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Amiram Eldar, Table of n, a(n) for n = 1..377 (terms 1..100 from T. D. Noe)
Thomas Baruchel and Carsten Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
Russell Easterly, Product Bases A Million Ways to Count [Archive link]
OEIS Wiki, LCM numeral system
Mike Winkler, Table of n, a(n), p for n = 3..200, 2013.
Index entries for sequences related to lcm's
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FORMULA
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a(n) = A003418(A000961(n)).
a(n) = A208768(n) + 1. - Reinhard Zumkeller, Mar 01 2012
Partial products of A014963. - Charles R Greathouse IV, Apr 28 2014
Sum_{n>=1} 1/a(n) = A064890. - Amiram Eldar, Nov 16 2020
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EXAMPLE
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lcm[1,...,n] is 2520 for n=9 and 10. The smallest such n's are always prime powers, where A003418 jumps.
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MATHEMATICA
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f[n_] := LCM @@ Range@ n; Union@ Array[f, 41] (* Robert G. Wilson v, Jul 11 2011 *)
Join[{1}, LCM@@Range[#]&/@Select[Range[50], PrimePowerQ]] (* Harvey P. Dale, Feb 06 2020 *)
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PROG
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(Haskell)
a051451 n = a051451_list !! (n-1)
a051451_list = scanl1 lcm a000961_list
-- Reinhard Zumkeller, Mar 01 2012
(PARI) do(lim)=my(v=primes(primepi(lim)), u=List([1])); forprime(p=2, sqrtint(lim\1), for(e=2, log(lim+.5)\log(p), listput(u, p^e))); v=vecsort(concat(v, Vec(u))); for(i=2, #v, v[i]=lcm(v[i], v[i-1])); v \\ Charles R Greathouse IV, Nov 20 2012
(PARI) {lim=100; n=1; i=1; j=1; until(n==lim, until(a!=j, a=lcm(j, i+1); i++; ); j=a; n++; print(n" "a); ); } \\ Mike Winkler, Sep 07 2013
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CROSSREFS
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Cf. A000961, A003418, A025473, A049536, A049537, A064890.
Sequence in context: A068625 A162935 A328459 * A090951 A168262 A085819
Adjacent sequences: A051448 A051449 A051450 * A051452 A051453 A051454
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Labos Elemer, Dec 11 1999
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EXTENSIONS
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Minor edits by Ray Chandler, Jan 16 2009
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STATUS
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approved
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