A176379 Smallest primes p which give a prime iterated by f(p,k) := 2 * p + prime(k+1) for at least two steps (k = 1, 2, ...).

2, 7, 2, 31, 2, 7, 11, 7, 19, 5, 5, 19, 2, 13, 13, 61, 11, 17, 61, 5, 5, 7, 139, 5, 19, 2, 103, 29, 7, 2, 109, 7, 59, 31, 41, 5, 5, 127, 13, 31, 5, 109, 2, 7, 41, 11, 2, 7, 101, 67

Offset

1,1

Comments

Such p are generalized Cunningham primes: prime numbers p(1), ..., p(n):

p(i+1) = a * p(i) + b (1 <= i < n) for fixed coprime integers a and b

Terms of this sequence the special case: a = 2, b = prime(k+1) (k = 1, 2, ...), n = 3

p, f(p) = 2 * p + prime(k+1), q = f(f(p)) = 4 * p + 3 * prime(k+1) to be primes

k = 0 is omitted as f(p,0) = 2 * p + prime(1) = 2 * (p+1) is even, only f(0,0) = 2 is prime(1)

List of (p,f(p),q):

(2,7,17) (7,19,43) (2,11,29) (31,73,157) (2,17,47)

(7,31,79) (11,41,101) (7,37,97) (19,67,163) (5,41,113)

(5,47,131) (19,79,199) (2,47,137) (13,73,193) (13,79,211)

(61,181,421) (11,83,227) (17,101,269) (61,193,457) (5,83,239)

(5,89,257) (7,97,277) (139,367,823) (5,107,311) (19,139,379)

(2,107,317) (103,313,733) (29,167,443) (7,127,367) (2,131,389)

(109,349,829) (7,151,439) (59,257,653) (31,211,571) (41,233,617)

(5,167,491) (5,173,509) (127,421,1009) (13,199,571) (31,241,661)

(5,191,563) (109,409,1009) (2,197,587) (7,211,619) (41,281,761)

(11,233,677) (2,227,677) (7,241,709) (101,431,1091) (67,367,967)

References

Joe Buhler: Algorithmic Number Theory: Third International Symposium, ANTS-III, Springer New York, 1998

R. K. Guy: Unsolved problems in number theory, Springer-Verlag, New York, 1994

Paulo Ribenboim: Die Welt der Primzahlen. Geheimnisse und Rekorde, Springer-Verlag GmbH & Co. KG, 2006

Links

Table of n, a(n) for n=1..50.

Example

k=1, prime(1+1) = 3: 2 * 2 + 3 = 7, 2 * 7 + 3 = 17, 2 is first term
k=2: 2 * 7 + 5 = 19, 2 * 19 + 5 = 43, 7 is 2nd term
k=3: 2 * 2 + 7 = 11, 2 * 11 + 7 = 29, 2 is 3rd term

Crossrefs

A000040, A005602, A059761, A059762, A110024

Sequence in context: A282260 A217202 A100489 * A282454 A282655 A282803

Adjacent sequences: A176376 A176377 A176378 * A176380 A176381 A176382

Keyword

base,nonn,uned

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 16 2010

Status

approved