login
a(1)=1, a(2)=2; thereafter, a(n) is the smallest number m not yet in the sequence such that every prime that divides a(n-1) also divides m.
45

%I #77 Jun 09 2024 13:21:02

%S 1,2,4,6,12,18,24,30,60,90,120,150,180,210,420,630,840,1050,1260,1470,

%T 1680,1890,2100,2310,4620,6930,9240,11550,13860,16170,18480,20790,

%U 23100,25410,27720,30030,60060,90090,120120,150150,180180,210210

%N a(1)=1, a(2)=2; thereafter, a(n) is the smallest number m not yet in the sequence such that every prime that divides a(n-1) also divides m.

%C Also, numbers k at which k / (phi(k) + 1) increases.

%C Except for the initial 1, this sequence is a primorial (A002110) followed by its multiples until the next primorial, then the multiples of that primorial and so on. - Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Dec 28 2006

%C a(1)=1, a(2)=2. For n >= 3, a(n) is the smallest integer > a(n-1) that is divisible by every prime which divides lcm(a(1), a(2), a(3), ..., a(n)). - _Leroy Quet_, Feb 23 2010

%C Numbers n for which A053589(n) = A260188(n), thus numbers with only one nonzero digit when written in primorial base A049345. - _Antti Karttunen_, Aug 30 2016

%C Lexicographically earliest infinite sequence of distinct positive numbers with property that every prime that divides a(n-1) also divides a(n). - _N. J. A. Sloane_, Apr 08 2022

%H Trey Deitch, <a href="/A060735/b060735.txt">Table of n, a(n) for n = 1..20000</a> (terms 1..5000 from Enrique Pérez Herrero)

%H Michel Planat, <a href="http://arxiv.org/abs/1010.3239">Riemann hypothesis from the Dedekind psi function</a>, arXiv:1010.3239 [math.GM], 2010.

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%F a(1) = 1, a(n) = a(n-1) + rad(a(n-1)) with rad=A007947, squarefree kernel. - _Reinhard Zumkeller_, Apr 10 2006

%F a(A101301(n)+1) = A002110(n). - _Enrique Pérez Herrero_, Jun 10 2012

%e After a(2)=2 the next term must be even, so a(3)=4.

%e Then a(4) must be even so a(4) = 6.

%e Now a(5) must be a multiple of 2*3=6, so a(5)=12.

%e Then a(6)=18, a(7)=24, a(8)=30.

%e Now a(9) must be a multiple of 2*3*5 = 30, so a(9)=60. And so on.

%p seq(seq(k*mul(ithprime(i),i=1..n-1),k=1..ithprime(n)-1),n=1..10); # _Vladeta Jovovic_, Apr 08 2004

%p a := proc(n) option remember; if n=1 then return 1 fi; a(n-1);

%p % + convert(numtheory:-factorset(%), `*`) end:

%p seq(a(n), n=1..42); # after Zumkeller, _Peter Luschny_, Aug 30 2016

%t a = 0; Do[ b = n/(EulerPhi[ n ] + 1); If[ b > a, a = b; Print[ n ] ], {n, 1, 10^6} ]

%t f[n_] := Range[Prime[n + 1] - 1] Times @@ Prime@ Range@ n; Array[f, 7, 0] // Flatten (* _Robert G. Wilson v_, Jul 22 2015 *)

%o (PARI) first(n)=my(v=vector(n),k=1,p=1,P=1); v[1]=1; for(i=2,n, v[i]=P*k++; if(k>p && isprime(k), p=k; P=v[i]; k=1)); v \\ _Charles R Greathouse IV_, Jul 22 2015

%o (PARI) is_A060735(n,P=1)={forprime(p=2,,n>(P*=p)||return(1);n%P&&return)} \\ _M. F. Hasler_, Mar 14 2017

%o (Python)

%o from functools import cache;

%o from sympy import primefactors, prod

%o @cache

%o def a(n): return 1 if n == 0 else a(n-1) + prod(primefactors(a(n-1)))

%o print([a(n) for n in range(42)]) # _Trey Deitch_, Jun 08 2024

%Y Cf. A000010, A002110, A049345, A055719, A101301, A053589, A260188.

%Y Indices of ones in A276157 and A267263.

%K nonn

%O 1,2

%A _Robert G. Wilson v_, Apr 23 2001

%E Definition corrected by _Franklin T. Adams-Watters_, Apr 16 2009

%E Simpler definition, comments, examples from _N. J. A. Sloane_, Apr 08 2022