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

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, 29, 37, 33, 31, 37, 33, 49, 37, 37, 41, 49, 39, 41, 43, 49, 43, 49, 45, 49, 55, 49, 49, 65, 57, 61, 55, 57, 55, 73, 61, 65, 61, 61, 61, 73, 63, 65, 73, 97, 71, 73, 69

Offset: 1

Author

Cody M. Haderlie, Feb 22 2016

Keywords

• nonn
• easy
• hear

Comments

a(n) is odd except when n=2.

Initially, a(n) is frequently a square or a prime.

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.

Examples

			For n=18765, a(n)=18901.
For n=196, a(n)=225 (225 is a square).
For n=103156, a(n)=105673 (105673 is prime).
		

Links

• Cody M. Haderlie, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006530, A052126, A270807.

Programs

Mathematica

Table[n+n/FactorInteger[n][[-1,1]]+1,{n,100}]

PARI

gpf(n)=if(n>1, my(f=factor(n)[,1]); f[#f], 1)
a(n)=n + n/gpf(n) + 1 \\ Charles R Greathouse IV, Feb 22 2016

Formula

a(n) = n + n/A006530(n) + 1.

a(n) = n + A052126(n) + 1.

a(p) = p+2 for p prime.