A051451 a(n) = lcm{ 1,2,...,x } where x is the n-th prime power (A000961).

1, 2, 6, 12, 60, 420, 840, 2520, 27720, 360360, 720720, 12252240, 232792560, 5354228880, 26771144400, 80313433200, 2329089562800, 72201776446800, 144403552893600, 5342931457063200, 219060189739591200, 9419588158802421600, 442720643463713815200

Offset: 1

Author

Labos Elemer, Dec 11 1999

Keywords

• nonn
• nice
• easy

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.

Conjecture: Every term = A001013(j)*A129912(k) for some j,k. (End)

Examples

			lcm[1,...,n] is 2520 for n=9 and 10. The smallest such n's are always prime powers, where A003418 jumps.
		

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.
• Index entries for sequences related to lcm's

Crossrefs

Cf. A000961, A003418, A025473, A049536, A049537, A064890, A025487.

Programs

Haskell

a051451 n = a051451_list !! (n-1)
a051451_list = scanl1 lcm a000961_list
-- Reinhard Zumkeller, Mar 01 2012

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 *)

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

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

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

Formula

a(n) = A003418(A000961(n)).

a(n) = A208768(n) + 1. - Reinhard Zumkeller, Mar 01 2012

Sum_{n>=1} 1/a(n) = A064890. - Amiram Eldar, Nov 16 2020

Extensions

Minor edits by Ray Chandler, Jan 16 2009