A208768 -id:A208768 - cp's OEIS frontend

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

Labos Elemer, Dec 11 1999

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)

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

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

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

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

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 *)

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

{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

x=1;for(i=1,100,if(omega(i)==1,x*=factor(i)[1,1])) \\ Florian Baur, Apr 11 2022

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

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

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

Minor edits by Ray Chandler, Jan 16 2009

A070198 Smallest nonnegative number m such that m == i (mod i+1) for all 1 <= i <= n.

Original entry on oeis.org

0, 1, 5, 11, 59, 59, 419, 839, 2519, 2519, 27719, 27719, 360359, 360359, 360359, 720719, 12252239, 12252239, 232792559, 232792559, 232792559, 232792559, 5354228879, 5354228879, 26771144399, 26771144399, 80313433199, 80313433199

Offset: 0

Benoit Cloitre, May 06 2002

Also, smallest k such that, for 0 <= i < n, i+1 divides k-i.
Suggested by Chinese Remainder Theorem. This sequence can generate others: smallest b(n) such that b(n) == i (mod (i+2)), 1 <= i <= n, gives b(1)=1 and b(n) = a(n+1)-1 for n > 1; smallest c(n) such that c(n) == i (mod (i+3)), 1 <= i <= n, gives c(1)=1, c(2)=17 and c(n) = a(n+2) - 2 for n > 2; smallest d(n) such that c(n) == i (mod (i+4)), 1 <= i <= n, gives d(1)=1, d(2)=26, d(3)=206 and d(n) = a(n+3) - 3 for n > 3, etc.
A208768(n) occurs A057820(n) times. - Reinhard Zumkeller, Mar 01 2012
From Kival Ngaokrajang, Oct 10 2013: (Start)
A070198(n-1) is m such that max(Sum_{i=1..n} m (mod i)) = A000217(n-1).
Example for n = 3:
  m\i = 1  2  3  sum
  1     0  1  1   2
  2     0  0  2   2
  3     0  1  0   1
  4     0  0  1   1
  5     0  1  2   3 <--max remainder sum = 3 = A000217(2)
  6     0  0  0   0  first occurs at m = 5 = A070198(2)
(End)

			a(3) = 11 because 11 == 1 (mod 2), 11 == 2 (mod 3) and 11 == 3 (mod 4).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Chinese Remainder Theorem
Wikipedia, Chinese remainder theorem
Index entries for sequences related to lcm's

Cf. A053664, A072562.
Cf. A057825 (indices of primes). - R. J. Mathar, Jan 14 2009
Cf. A116151. - Zak Seidov, Mar 11 2014

a070198 n = a070198_list !! n
a070198_list = map (subtract 1) $ scanl lcm 1 [2..]
-- Reinhard Zumkeller, Mar 01 2012

[Exponent(SymmetricGroup(n))-1 : n in [1..30]]; /* Vincenzo Librandi, Oct 31 2014 - after Arkadiusz Wesolowski in A003418 */

seq(ilcm($1..n) - 1, n=1..100); # Robert Israel, Nov 03 2014

f[n_] := ChineseRemainder[ Range[0, n - 1], Range[n]]; Array[f, 28] (* or *)
f[n_] := LCM @@ Range@ n - 1; Array[f, 28] (* Robert G. Wilson v, Oct 30 2014 *)

from math import lcm
def A070198(n): return lcm(*range(1,n+2))-1 # Chai Wah Wu, May 02 2023

a(n) = lcm(1, 2, 3, ..., n+1) - 1 = A003418(n+1) - 1.

Edited by N. J. A. Sloane, Nov 18 2007, at the suggestion of Max Alekseyev

A051452 a(n) = 1 + lcm(1..k) where k is the n-th prime power A000961(n).

Original entry on oeis.org

2, 3, 7, 13, 61, 421, 841, 2521, 27721, 360361, 720721, 12252241, 232792561, 5354228881, 26771144401, 80313433201, 2329089562801, 72201776446801, 144403552893601, 5342931457063201, 219060189739591201

Offset: 1

Labos Elemer

nonn

From Daniel Forgues, Apr 27 2014: (Start)
Factorizations:
  2, 3, 7, 13, 61, 421 are primes;
  841 = 29^2;
  2521 is prime;
  27721 = 19*1459, 360361 = 89*4049, 720721 = 71*10151,
  12252241 = 1693*7237;
  232792561 is prime;
  5354228881 = 6659*804059;
  26771144401 is prime;
  80313433201 = 331*11239*21589, 2329089562801 = 101*271*2311*36821;
  72201776446801 is prime.
Very likely contains an infinite number of primes (see A049536). (End)

1 + A003418(A000961(n)), corresponds to distinct values of 1 + A003418.

Cf. A049536, A049537, A051454, A208768.

print1(2);t=1;for(n=2,100,if(t%n, t=lcm(t,n); print1(", "t+1))) \\ Charles R Greathouse IV, Jan 04 2013

from math import prod
from sympy import primepi, integer_nthroot, integer_log, primerange
def A051452(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 1+prod(p**integer_log(m, p)[0] for p in primerange(m+1)) # Chai Wah Wu, Aug 15 2024

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

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A070198 Smallest nonnegative number m such that m == i (mod i+1) for all 1 <= i <= n.

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A051452 a(n) = 1 + lcm(1..k) where k is the n-th prime power A000961(n).

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