cp's OEIS Frontend

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.

%I A060735 #80 Nov 14 2024 08:22:24
%S A060735 1,2,4,6,12,18,24,30,60,90,120,150,180,210,420,630,840,1050,1260,1470,
%T A060735 1680,1890,2100,2310,4620,6930,9240,11550,13860,16170,18480,20790,
%U A060735 23100,25410,27720,30030,60060,90090,120120,150150,180180,210210
%N 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.
%C A060735 Also, numbers k at which k / (phi(k) + 1) increases.
%C A060735 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 A060735 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 A060735 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 A060735 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 A060735 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 A060735 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 A060735 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A060735 a(1) = 1, a(n) = a(n-1) + rad(a(n-1)) with rad=A007947, squarefree kernel. - _Reinhard Zumkeller_, Apr 10 2006
%F A060735 a(A101301(n)+1) = A002110(n). - _Enrique Pérez Herrero_, Jun 10 2012
%F A060735 a(n) = 1 + A343048(n). - _Antti Karttunen_, Nov 14 2024
%e A060735 After a(2)=2 the next term must be even, so a(3)=4.
%e A060735 Then a(4) must be even so a(4) = 6.
%e A060735 Now a(5) must be a multiple of 2*3=6, so a(5)=12.
%e A060735 Then a(6)=18, a(7)=24, a(8)=30.
%e A060735 Now a(9) must be a multiple of 2*3*5 = 30, so a(9)=60. And so on.
%p A060735 seq(seq(k*mul(ithprime(i),i=1..n-1),k=1..ithprime(n)-1),n=1..10); # _Vladeta Jovovic_, Apr 08 2004
%p A060735 a := proc(n) option remember; if n=1 then return 1 fi; a(n-1);
%p A060735 % + convert(numtheory:-factorset(%), `*`) end:
%p A060735 seq(a(n), n=1..42); # after Zumkeller, _Peter Luschny_, Aug 30 2016
%t A060735 a = 0; Do[ b = n/(EulerPhi[ n ] + 1); If[ b > a, a = b; Print[ n ] ], {n, 1, 10^6} ]
%t A060735 f[n_] := Range[Prime[n + 1] - 1] Times @@ Prime@ Range@ n;  Array[f, 7, 0] // Flatten (* _Robert G. Wilson v_, Jul 22 2015 *)
%o A060735 (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 A060735 (PARI) is_A060735(n,P=1)={forprime(p=2,,n>(P*=p)||return(1);n%P&&return)} \\ _M. F. Hasler_, Mar 14 2017
%o A060735 (Python)
%o A060735 from functools import cache;
%o A060735 from sympy import primefactors, prod
%o A060735 @cache
%o A060735 def a(n): return 1 if n == 0 else a(n-1) + prod(primefactors(a(n-1)))
%o A060735 print([a(n) for n in range(42)]) # _Trey Deitch_, Jun 08 2024
%Y A060735 Cf. A000010, A002110, A049345, A055719, A101301, A053589, A260188.
%Y A060735 Indices of ones in A276157 and A267263.
%Y A060735 One more than A343048.
%K A060735 nonn
%O A060735 1,2
%A A060735 _Robert G. Wilson v_, Apr 23 2001
%E A060735 Definition corrected by _Franklin T. Adams-Watters_, Apr 16 2009
%E A060735 Simpler definition, comments, examples from _N. J. A. Sloane_, Apr 08 2022