cp's OEIS Frontend

A047836 "Nullwertzahlen" (or "inverse prime numbers"): n=p1*p2*p3*p4*p5*...*pk, where pi are primes with p1 <= p2 <= p3 <= p4 ...; then p1 = 2 and p1*p2*...*pi >= p(i+1) for all i < k.

%I A047836 #59 Apr 03 2023 18:17:32
%S A047836 2,4,8,12,16,24,32,36,40,48,56,60,64,72,80,84,96,108,112,120,128,132,
%T A047836 144,160,168,176,180,192,200,208,216,224,240,252,256,264,280,288,300,
%U A047836 312,320,324,336,352,360,384,392,396,400,408,416,420,432,440,448
%N A047836 "Nullwertzahlen" (or "inverse prime numbers"): n=p1*p2*p3*p4*p5*...*pk, where pi are primes with p1 <= p2 <= p3 <= p4 ...; then p1 = 2 and p1*p2*...*pi >= p(i+1) for all i < k.
%C A047836 Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. The "value" of n (A047988) is the smallest cost to reach 2. Sequence gives numbers with value 0.
%C A047836 a(n) is also the length of the largest Dyck path of the symmetric representation of sigma of the n-th number whose symmetric representation of sigma has only one part. For an illustration see A317305. (Cf. A237593.) - _Omar E. Pol_, Aug 25 2018
%C A047836 This sequence can be defined equivalently as the increasing terms of the set containing 2 and all the integers such that if n is in the set, then all m * n are in the set for all m <= n. - _Giuseppe Melfi_, Oct 21 2019
%C A047836 The subsequence giving the largest term with k prime factors (k >= 1) starts 2, 4, 12, 132, 17292, 298995972, ... . - _Peter Munn_, Jun 04 2020
%H A047836 T. D. Noe, <a href="/A047836/b047836.txt">Table of n, a(n) for n = 1..1000</a>
%H A047836 Thomas Kantke, <a href="http://www.spektrum.de/magazin/das-spiel-minimum-und-die-zerlegung-natuerlicher-zahlen/820793">Das Spiel Minimum und die Zerlegung natürlicher Zahlen</a>, Spektrum der Wissenschaft, No. 4, 1993, pp. 11-13.
%H A047836 Andreas Weingartner, <a href="https://arxiv.org/abs/2303.16819">Uniform distribution of alpha*n modulo one for a family of integer sequences</a>, arXiv:2303.16819 [math.NT], 2023.
%F A047836 a(n) = 2 * A174973(n). - _Reinhard Zumkeller_, Sep 28 2011
%F A047836 The number of terms <= x is c*x/log(x) + O(x/(log(x))^2), where c = 0.612415..., and a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 1.63287... This follows from the formula just above. - _Andreas Weingartner_, Jun 30 2021
%e A047836 Starting at 24 we divide by 3, 2, then 2, reaching 2.
%t A047836 nMax = 100; A174973 = Select[Range[10*nMax], AllTrue[Rest[dd = Divisors[#]] / Most[dd], Function[r, r <= 2]]&]; a[n_] := 2*A174973[[n]]; Array[a, nMax] (* _Jean-François Alcover_, Nov 10 2016, after _Reinhard Zumkeller_ *)
%o A047836 (Haskell)
%o A047836 import Data.List.Ordered (union)
%o A047836 a047836 n = a047836_list !! (n-1)
%o A047836 a047836_list = f [2] where
%o A047836    f (x:xs) = x : f (xs `union` map (x *) [2..x])
%o A047836 -- _Reinhard Zumkeller_, Jun 25 2015, Sep 28 2011
%Y A047836 Cf. A047984, A047985, A047986, A047987, A047988, A052287, A237593, A317305.
%K A047836 nonn,nice,easy
%O A047836 1,1
%A A047836 Thomas Kantke (bytes.more(AT)ibm.net)
%E A047836 More terms from _David W. Wilson_