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A269304 a(n) = n + n/gpf(n) + 1, where gpf(n) is the greatest prime factor of n or 1 if n = 1.

%I A269304 #107 Apr 05 2016 17:49:29
%S A269304 3,4,5,7,7,9,9,13,13,13,13,17,15,17,19,25,19,25,21,25,25,25,25,33,31,
%T A269304 29,37,33,31,37,33,49,37,37,41,49,39,41,43,49,43,49,45,49,55,49,49,65,
%U A269304 57,61,55,57,55,73,61,65,61,61,61,73,63,65,73,97,71,73,69
%N A269304 a(n) = n + n/gpf(n) + 1, where gpf(n) is the greatest prime factor of n or 1 if n = 1.
%C A269304 a(n) is odd except when n=2.
%C A269304 Initially, a(n) is frequently a square or a prime.
%C A269304 It is conjectured that any two sequences generated with a(n)=a(n-1)+a(n-1)/gpf(a(n-1))+1 and any initial value >=1 will have a finite number of non-shared terms and an infinite number of shared terms after one initial shared term (see A270807). Example: For a(1)=314, the sequence generated is 314, 317, 319, 331, 333, 343, 393, 397, 399, 421, 423, 433, ...; for a(1)=97, the sequence generated is 97, 99, 109, 111, 115, 121, 133, 141, 145, 151, 153, 163, 165, 181, 183, 187, 199, 201, 205, 211, 213, 217, 225, 271, 273, 295, 301, 309, 313, 315, 361, 381, 385, 421, 423, 433, ...; these sequences have respectively 9 and 33 terms not shared with the other until both reach 421; the following terms of both sequences are identical.
%H A269304 Cody M. Haderlie, <a href="/A269304/b269304.txt">Table of n, a(n) for n = 1..10000</a>
%F A269304 a(n) = n + n/A006530(n) + 1.
%F A269304 a(n) = n + A052126(n) + 1.
%F A269304 a(p) = p+2 for p prime.
%e A269304 For n=18765, a(n)=18901.
%e A269304 For n=196, a(n)=225 (225 is a square).
%e A269304 For n=103156, a(n)=105673 (105673 is prime).
%t A269304 Table[n+n/FactorInteger[n][[-1,1]]+1,{n,100}]
%o A269304 (PARI) gpf(n)=if(n>1, my(f=factor(n)[,1]); f[#f], 1)
%o A269304 a(n)=n + n/gpf(n) + 1 \\ _Charles R Greathouse IV_, Feb 22 2016
%Y A269304 Cf. A006530, A052126, A270807.
%K A269304 nonn,easy,hear
%O A269304 1,1
%A A269304 _Cody M. Haderlie_, Feb 22 2016