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.

1, 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

Offset

1,2

Comments

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

Essentially the same as A036441(n) = a(n+1) and A180107(n) = a(n-1) (n > 1).

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

From David James Sycamore, Apr 27 2017: (Start)

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).

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.

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)

Links

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

Formula

a(A076274(n)) = A008578(n)^2 for all n.

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

Mathematica

NestList[#+FactorInteger[#][[-1, 1]]&, 1, 60] (* Harvey P. Dale, May 11 2015 *)

Prog

(PARI) print1(n=1); for(i=1, 199, print1(", "n+=A006530(n))) \\ M. F. Hasler, Apr 08 2015
(Haskell)
a076271 n = a076271_list !! (n-1)
a076271_list = iterate a070229 1  -- Reinhard Zumkeller, Nov 07 2015

Crossrefs

Cf. A036441, A076272(n) = a(n+1) - a(n).

See also A180107.

Cf. A070229.

Sequence in context: A130025 A278299 A145802 * A036441 A180107 A135146

Adjacent sequences: A076268 A076269 A076270 * A076272 A076273 A076274

Keyword

nonn

Author

Reinhard Zumkeller, Oct 04 2002

Extensions

Edited by M. F. Hasler, Apr 08 2015

Status

approved