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A036441 a(n+1) = next number having largest prime dividing a(n) as a factor, with a(1) = 2.

%I A036441 #39 Jul 02 2025 19:35:15
%S A036441 2,4,6,9,12,15,20,25,30,35,42,49,56,63,70,77,88,99,110,121,132,143,
%T A036441 156,169,182,195,208,221,238,255,272,289,306,323,342,361,380,399,418,
%U A036441 437,460,483,506,529,552,575,598,621,644,667,696,725,754,783,812,841,870
%N A036441 a(n+1) = next number having largest prime dividing a(n) as a factor, with a(1) = 2.
%C A036441 a(n) satisfies the following inequality: (1/4)*(n^2 + 3*n + 1) <= a(n) <= (1/4)*(n+2)^2. [Corrected by _M. F. Hasler_, Apr 08 2015]
%C A036441 The present sequence is the special case a(n) = a(2,n) with a more general a(m, n) := a(m, n-1) + gpf(a(m, n-1)), a(m, 1) := m, where gpf(x) := "greatest prime factor of x" = A006530(x). Also a(a(r,k), n) = a(r,n+k-1), for all n,k in N\{0} and all r in N\{0,1}; a(prime(k), n) = a(prime(i), n + prime(k) - prime(i)), for all k,i,n in N\{0}, with k >= i, n >= prime(k-1) and with prime(x) := x-th prime.
%C A036441 Essentially the same as A076271 and A180107, cf. formula.
%H A036441 T. D. Noe, <a href="/A036441/b036441.txt">Table of n, a(n) for n = 1..10000</a>
%F A036441 a(n) = p(m)*(n+2-p(m)), where p(k) is the k-th prime and m is the smallest index such that n+2 <= p(m) + p(m+1). - _Max Alekseyev_, Oct 21 2008
%F A036441 a(n) = A076271(n+1) = A180107(n+2). - _M. F. Hasler_, Apr 08 2015
%F A036441 a(n+1) = A070229(a(n)). - _Reinhard Zumkeller_, Nov 07 2015
%e A036441 a(2,2) = 4 because 2 + gpf(2) = 2 + 2 = 4;
%e A036441 a(2,3) = 6 because 4 + gpf(4) = 4 + 2 = 6.
%t A036441 f[n_]:=Last[First/@FactorInteger[n]];Join[{a=2},Table[a+=f[a],{n,2,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2011*)
%t A036441 NestList[#+FactorInteger[#][[-1,1]]&,2,60] (* _Harvey P. Dale_, Dec 02 2012 *)
%o A036441 (Haskell)
%o A036441 a036441 n = a036441_list !! (n-1)
%o A036441 a036441_list = tail a076271_list
%o A036441 -- _Reinhard Zumkeller_, Nov 08 2015, Nov 14 2011
%o A036441 (PARI) a(n)=(n+2-if(n\2+1<(p=nextprime(n\2+1))&&n+1<p+n=precprime(p-1),p=n,p))*p \\ _M. F. Hasler_, Apr 08 2015
%Y A036441 Cf. A006530. See A076271 and A180107 for other versions.
%Y A036441 Cf. A001043, A070229.
%Y A036441 Cf. A123581.
%Y A036441 Partial sums of A076973.
%K A036441 eigen,nice,nonn
%O A036441 1,1
%A A036441 Frederick Magata (frederick.magata(AT)uni-muenster.de)
%E A036441 Better description from _Reinhard Zumkeller_, Feb 04 2002
%E A036441 Edited by _M. F. Hasler_, Apr 08 2015