A051451 - OEIS

%I #140 Aug 19 2024 19:55:42

%S 1,2,6,12,60,420,840,2520,27720,360360,720720,12252240,232792560,

%T 5354228880,26771144400,80313433200,2329089562800,72201776446800,

%U 144403552893600,5342931457063200,219060189739591200,9419588158802421600,442720643463713815200

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

%C This sequence is the list of distinct terms in A003418.

%C 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

%C Partial products of A025473, prime roots of the prime powers.

%C 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

%C The prime powers give all values, and do so uniquely. (Other positive integers give repeated values.) - _Daniel Forgues_, Apr 28 2014

%C "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

%C 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

%C 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

%C From _Bill McEachen_, Apr 28 2023: (Start)

%C Every term belongs to A025487.

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

%H Amiram Eldar, <a href="/A051451/b051451.txt">Table of n, a(n) for n = 1..377</a> (terms 1..100 from T. D. Noe)

%H Thomas Baruchel and Carsten Elsner, <a href="http://arxiv.org/abs/1602.06445">On error sums formed by rational approximations with split denominators</a>, arXiv preprint arXiv:1602.06445 [math.NT], 2016.

%H Russell Easterly, <a href="https://web.archive.org/web/20151002183910/http://home.comcast.net/~logiclab/ProductBase.html">Product Bases A Million Ways to Count</a> [Archive link]

%H OEIS Wiki, <a href="/wiki/LCM_numeral_system">LCM numeral system</a>

%H Mike Winkler, <a href="https://web.archive.org/web/20160804121249/http://mikewinkler.co.nf/twinprimesconjecture2013.pdf">Table of n, a(n), p for n = 3..200</a>, 2013.

%H <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>

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

%F a(n) = A208768(n) + 1. - _Reinhard Zumkeller_, Mar 01 2012

%F Sum_{n>=1} 1/a(n) = A064890. - _Amiram Eldar_, Nov 16 2020

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

%t f[n_] := LCM @@ Range@ n; Union@ Array[f, 41] (* _Robert G. Wilson v_, Jul 11 2011 *)

%t Join[{1},LCM@@Range[#]&/@Select[Range[50],PrimePowerQ]] (* _Harvey P. Dale_, Feb 06 2020 *)

%o (Haskell)

%o a051451 n = a051451_list !! (n-1)

%o a051451_list = scanl1 lcm a000961_list

%o -- _Reinhard Zumkeller_, Mar 01 2012

%o (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

%o (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

%o (PARI) x=1;for(i=1,100,if(omega(i)==1,x*=factor(i)[1,1])) \\ _Florian Baur_, Apr 11 2022

%o (Sage)

%o def A051451_list(n):

%o     a = [ ]

%o     L = [1]

%o     for i in (1..n):

%o        a.append(i)

%o        if (is_prime_power(i) == 1):

%o            L.append(lcm(a))

%o     return(L)

%o A051451_list(42) # _Jani Melik_, Jul 07 2022

%o (Python)

%o from math import prod

%o from sympy import primepi, integer_nthroot, integer_log, primerange

%o def A051451(n):

%o     def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return prod(p**integer_log(m, p)[0] for p in primerange(m+1)) # _Chai Wah Wu_, Aug 15 2024

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

%K nonn,nice,easy

%O 1,2

%A _Labos Elemer_, Dec 11 1999

%E Minor edits by _Ray Chandler_, Jan 16 2009