|
| |
|
|
A113927
|
|
a(1)=1, and recursively a(n+1) is the smallest prime p of the form p = 2*a(n) + 5^k for some k>0.
|
|
0
|
|
|
|
1, 7, 19, 43, 211, 547, 4219, 8443, 17011, 34147, 71419, 142963, 1220989051, 3662681227, 19080811690579, 38161623381163, 76324467465451, 152648936884027, 305299094471179, 4656613483675581520483
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Note that last digits cycle 7, 9, 3, 1; 7, 9, 3, 1. Note that the exponent k of 5^k is always odd. This follows from taking this sequence mod 6.
Since the first prime value a(2) = 7 == 1 mod 6, all values a(n) thereafter are primes of the form 6*d+1. Hence a(n+1) = [2*(6*d+1) + 5^2] mod 6 == 12*d + 2 + 1 == 3 mod 6 and would be divisible by 3; a(n+1) = [2*(6*d+1) + 5^4] mod 6 == 12*d + 2 + 1 == 3 mod 6 and would be divisible by 3; and so for all even exponents.
In general, the (b,c,d) Jasinski-like positive power sequence is defined as follows: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for positive integer k. The (b,c,d) Jasinski-like nonnegative power sequence is defined: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for integer k. In this notation, A113824 is the (1,2,2) Jasinski-like nonnegative power sequence. A113914 is the (1,2,3) Jasinski-like positive power sequence, and this here the (1,2,5) Jasinski-like power sequence.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
a(1) = 1 by definition.
a(2) = 2*1 + 5^1 = 7.
a(3) = 2*7 + 5^1 = 19.
a(4) = 2*19 + 5^1 = 43.
a(5) = 2*43 + 5^3 = 211.
a(6) = 2*211 + 5^3 = 547.
a(7) = 2*547 + 5^5 = 4219.
a(13) = 2*142963 + 5^13 = 1220989051.
a(20) = 2*305299094471179 + 5^31 = 4656613483675581520483, where 31 is a record exponent.
a(22) = 2*9313226967351163119091 + 5^45 = 28421709449030461369547296941307 and 45 is the new record exponent.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
