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

2, 4, 6, 9, 12, 15, 20, 25, 30, 35, 42, 49, 56, 63, 70, 77, 88, 99, 110, 121, 132, 143, 156, 169, 182, 195, 208, 221, 238, 255, 272, 289, 306, 323, 342, 361, 380, 399, 418, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 696, 725, 754, 783, 812, 841, 870

Offset

1,1

Comments

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]

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.

Essentially the same as A076271 and A180107, cf. formula.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Formula

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

a(n) = A076271(n+1) = A180107(n+2). - M. F. Hasler, Apr 08 2015

a(n+1) = A070229(a(n)). - Reinhard Zumkeller, Nov 07 2015

Example

a(2,2) = 4 because 2 + gpf(2) = 2 + 2 = 4;
a(2,3) = 6 because 4 + gpf(4) = 4 + 2 = 6.

Mathematica

f[n_]:=Last[First/@FactorInteger[n]]; Join[{a=2}, Table[a+=f[a], {n, 2, 100}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011*)
NestList[#+FactorInteger[#][[-1, 1]]&, 2, 60] (* Harvey P. Dale, Dec 02 2012 *)

Prog

(Haskell)
a036441 n = a036441_list !! (n-1)
a036441_list = tail a076271_list
-- Reinhard Zumkeller, Nov 08 2015, Nov 14 2011
(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

Crossrefs

Cf. A006530. See A076271 and A180107 for other versions.

Cf. A001043, A070229.

Cf. A123581.

Partial sums of A076973.

Sequence in context: A278299 A145802 A076271 * A180107 A135146 A256956

Adjacent sequences: A036438 A036439 A036440 * A036442 A036443 A036444

Keyword

eigen,nice,nonn

Author

Frederick Magata (frederick.magata(AT)uni-muenster.de)

Extensions

Better description from Reinhard Zumkeller, Feb 04, 2002

Edited by M. F. Hasler, Apr 08 2015

Status

approved