Rowland's sequence

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Rowland's sequence, named after Eric Rowland, is defined by the recurrence

${\begin{aligned}a(1)&=7;\\a(n)&=a(n-1)+\gcd(n,\,a(n-1)),\quad n>1.\end{aligned}}$

A106108 Rowland's sequence: 

a (1) = 7

, then 

a (n) = a (n  −  1) + gcd (n, a (n  −  1)), n   >   1

.

 

{7, 8, 9, 10, 15, 18, 19, 20, 21, 22, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, ...}

Sequences

A132199 Rowland's prime-generating sequence: first differences of A106108. The difference between two consecutive terms (A132199) is a noncomposite number (either the unit 

1

, or a prime number).

 

{1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}

A137613 Omit the 

1

s from the sequence 

f  (n)  −  f  (n  −  1) = gcd (n, f  (n  −  1))

, where 

f  (1) = 7

. Rowland proves that each term is prime. He says it is likely that all odd primes occur.

 

{5, 3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, 5, 3, 941, 3, 7, 1889, 3, 3779, 3, 7559, 3, 13, 15131, 3, 53, 3, 7, 30323, 3, 60647, 3, 5, 3, 101, 3, ...}