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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.
13
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)
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 A375983
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Oct 04 2002
EXTENSIONS
Edited by M. F. Hasler, Apr 08 2015
STATUS
approved