cp's OEIS Frontend

A076271 a(1) = 1, a(2) = 2, and for n > 2, a(n) = a(n-1) + gpf(a(n-1)), where gpf = greatest prime factor = A006530.

%I A076271 #49 Feb 27 2020 17:59:26
%S A076271 1,2,4,6,9,12,15,20,25,30,35,42,49,56,63,70,77,88,99,110,121,132,143,
%T A076271 156,169,182,195,208,221,238,255,272,289,306,323,342,361,380,399,418,
%U A076271 437,460,483,506,529,552,575,598,621,644,667,696,725,754,783,812,841
%N A076271 a(1) = 1, a(2) = 2, and for n > 2, a(n) = a(n-1) + gpf(a(n-1)), where gpf = greatest prime factor = A006530.
%C A076271 a(n+1) is the smallest number such that the largest prime divisor of a(n) is the highest common factor of a(n) and a(n+1). - _Amarnath Murthy_, Oct 17 2002
%C A076271 Essentially the same as A036441(n) = a(n+1) and A180107(n) = a(n-1) (n > 1).
%C A076271 The equivalent sequence with A020639 = spf instead of A006530 = gpf begins a(1) = 1, a(2) = 2, and from then on we get all even numbers: a(n) = a(2) + 2*(n-2), n > 1. - _M. F. Hasler_, Apr 08 2015
%C A076271 From _David James Sycamore_, Apr 27 2017: (Start)
%C A076271 The sequence contains only one prime; a(2)=2, all other terms (excluding a(1)=1) being composite, since if a(n) for some n > 2 is assumed to be the first prime after 2, then a(n) = a(n-1) + gpf(a(n-1))= m*q+q = q*(m+1) for some integer m > 1 and some prime q. This number is composite; contradiction. Terms after a(3)=4 alternate between even and odd values since each is created by addition of a prime (odd term).
%C A076271 All terms a(n) arise as consecutive multiples of consecutive primes occurring in their natural ascending order, 2,3,5,7.... (A000040). The number of (consecutive) terms which arise as multiples of p(n)= A000040(n) is 1 + p(n+1)- p(n-1), namely n-th term of the sequence: 2,4,5,7,7,7,7,7,11, etc. Example: Number of multiples of 17, the 7th prime, is 1+p(8)-p(6) = 1+19-13 = 7.
%C A076271 For any pair of consecutive primes, p,q (p < q) a(p+q-1) = p*q, the (semiprime) term where multiples of p end and multiples of q start. Example a(7+11-1) = a(17) = 77 = 11*7, the last multiple of 7 and first multiple of 11. Every string of multiples of prime p contains the term p^2, located at a(2*p-1). E.g.: a(3)=4, a(5)=9, a(9)=25. (End)
%H A076271 T. D. Noe, <a href="/A076271/b076271.txt">Table of n, a(n) for n = 1..10000</a>
%F A076271 a(A076274(n)) = A008578(n)^2 for all n.
%F A076271 a(n+1) = A070229(a(n)). - _Reinhard Zumkeller_, Nov 07 2015
%t A076271 NestList[#+FactorInteger[#][[-1,1]]&,1,60] (* _Harvey P. Dale_, May 11 2015 *)
%o A076271 (PARI) print1(n=1);for(i=1,199,print1(","n+=A006530(n))) \\ _M. F. Hasler_, Apr 08 2015
%o A076271 (Haskell)
%o A076271 a076271 n = a076271_list !! (n-1)
%o A076271 a076271_list = iterate a070229 1  -- _Reinhard Zumkeller_, Nov 07 2015
%Y A076271 Cf. A036441, A076272(n) = a(n+1) - a(n).
%Y A076271 See also A180107.
%Y A076271 Cf. A070229.
%K A076271 nonn
%O A076271 1,2
%A A076271 _Reinhard Zumkeller_, Oct 04 2002
%E A076271 Edited by _M. F. Hasler_, Apr 08 2015