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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

Table of n, a(n) for n=1..20.

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

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824, A113914.

Sequence in context: A155351 A155430 A155247 * A155399 A330854 A268926

Adjacent sequences:  A113924 A113925 A113926 * A113928 A113929 A113930

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Jan 30 2006

STATUS

approved

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Last modified June 21 10:02 EDT 2021. Contains 345358 sequences. (Running on oeis4.)