A060735 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.

1, 2, 4, 6, 12, 18, 24, 30, 60, 90, 120, 150, 180, 210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 4620, 6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720, 30030, 60060, 90090, 120120, 150150, 180180, 210210

Offset: 1

Author

Robert G. Wilson v, Apr 23 2001

Keywords

• nonn

Comments

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

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

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

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

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

Examples

			After a(2)=2 the next term must be even, so a(3)=4.
Then a(4) must be even so a(4) = 6.
Now a(5) must be a multiple of 2*3=6, so a(5)=12.
Then a(6)=18, a(7)=24, a(8)=30.
Now a(9) must be a multiple of 2*3*5 = 30, so a(9)=60. And so on.
		

Links

• Trey Deitch, Table of n, a(n) for n = 1..20000 (terms 1..5000 from Enrique Pérez Herrero)
• Michel Planat, Riemann hypothesis from the Dedekind psi function, arXiv:1010.3239 [math.GM], 2010.
• Index entries for sequences related to primorial base

Crossrefs

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

Indices of ones in A276157 and A267263.

One more than A343048.

Programs

Maple

seq(seq(k*mul(ithprime(i),i=1..n-1),k=1..ithprime(n)-1),n=1..10); # Vladeta Jovovic, Apr 08 2004
a := proc(n) option remember; if n=1 then return 1 fi; a(n-1);
% + convert(numtheory:-factorset(%), `*`) end:
seq(a(n), n=1..42); # after Zumkeller, Peter Luschny, Aug 30 2016

Mathematica

a = 0; Do[ b = n/(EulerPhi[ n ] + 1); If[ b > a, a = b; Print[ n ] ], {n, 1, 10^6} ]
f[n_] := Range[Prime[n + 1] - 1] Times @@ Prime@ Range@ n;  Array[f, 7, 0] // Flatten (* Robert G. Wilson v, Jul 22 2015 *)

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

PARI

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

Python

from functools import cache;
from sympy import primefactors, prod
@cache
def a(n): return 1 if n == 0 else a(n-1) + prod(primefactors(a(n-1)))
print([a(n) for n in range(42)]) # Trey Deitch, Jun 08 2024

Formula

a(1) = 1, a(n) = a(n-1) + rad(a(n-1)) with rad=A007947, squarefree kernel. - Reinhard Zumkeller, Apr 10 2006

a(A101301(n)+1) = A002110(n). - Enrique Pérez Herrero, Jun 10 2012

a(n) = 1 + A343048(n). - Antti Karttunen, Nov 14 2024

Extensions

Definition corrected by Franklin T. Adams-Watters, Apr 16 2009

Simpler definition, comments, examples from N. J. A. Sloane, Apr 08 2022