A060735 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

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

Indices of ones in A276157 and A267263.

One more than A343048.

Sequence in context: A175305 A342702 A171923 * A181416 A225566 A273009