OFFSET
1,2
COMMENTS
This sequence is the list of distinct terms in 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
Every term of this sequence is deeply composite (A095848). Moreover, the terms of this sequence are the "special deeply composite numbers", in analogy to the special highly composite numbers (A106037). A special highly composite number is a highly composite number (A002182) that divides every larger highly composite number. In the same fashion, the deeply composite numbers that divide every larger deeply composite number are just the terms of this sequence. This follows from the formula for deeply composite numbers. - Hal M. Switkay, Jun 08 2021
From Bill McEachen, Apr 28 2023: (Start)
Every term belongs to A025487.
LINKS
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.
FORMULA
a(n) = A208768(n) + 1. - Reinhard Zumkeller, Mar 01 2012
Sum_{n>=1} 1/a(n) = A064890. - Amiram Eldar, Nov 16 2020
EXAMPLE
lcm[1,...,n] is 2520 for n=9 and 10. The smallest such n's are always prime powers, where A003418 jumps.
MATHEMATICA
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 *)
PROG
(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
(PARI) x=1; for(i=1, 100, if(omega(i)==1, x*=factor(i)[1, 1])) \\ Florian Baur, Apr 11 2022
(Sage)
def A051451_list(n):
a = [ ]
L = [1]
for i in (1..n):
a.append(i)
if (is_prime_power(i) == 1):
L.append(lcm(a))
return(L)
A051451_list(42) # Jani Melik, Jul 07 2022
(Python)
from math import prod
from sympy import primepi, integer_nthroot, integer_log, primerange
def A051451(n):
def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length())))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
return prod(p**integer_log(m, p)[0] for p in primerange(m+1)) # Chai Wah Wu, Aug 15 2024
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
Labos Elemer, Dec 11 1999
EXTENSIONS
Minor edits by Ray Chandler, Jan 16 2009
STATUS
approved