A308193 - OEIS

%I #45 Jan 16 2023 18:57:50

%S 5,6,7,10,16,17,29,53,101,197,389,773,1542,3079,6154,12304,24604,

%T 36901,73798,147592,295180,295517,591030,1182056,1574849,3149694,

%U 4728211,6299383,12598762,25197520,25197533,50395062,100790119,201580234,403160464,806320924,1232145821,2464291638

%N Indices of records in A308190.

%C For terms a(1) through a(16), with one exception, 2*a(n) - a(n+1) is either 4 or 5. Does this pattern continue, and if so, why?

%C From _Chai Wah Wu_, Jun 14 2019: (Start)

%C The pattern does not continue. a(17) = 24604, a(18) = 36901.

%C Theorem:

%C 1. All terms are even or prime.

%C 2. If a(n+1) is even, then 2*a(n)-a(n+1) = 4.

%C 3. a(n+1) <= 2*(a(n)-2).

%C Proof: If a(n+1) = x is even, then A111234(x) = 2+x/2 = y. If we assume that x >= 6, then y < x. Thus A308190(x) = A308190(y)+1, i.e., a(n) <= y. If a(n) < y, then A308190(2*(a(n)-2)) = A308190(a(n)) + 1.

%C Since a(n) is a record value, this means that the next record value is at most at 2*(a(n)-2), i.e., 2*(a(n)-2) < x = a(n+1), a contradiction.

%C Thus we have shown that if a(n+1) is even, then 2*a(n) = a(n+1)+4.

%C If a(n+1) = x is an odd composite with smallest prime factor p > 2, then A308190(x) = A308190(y)+1 where y = p+x/p. On the other hand, A308190(2*(y-2)) = A308190(y)+1. Since 2*(y-2) < x, this contradicts the fact that a(n+1) = x is a record value.

%C (End)

%H Chai Wah Wu, <a href="/A308193/b308193.txt">Table of n, a(n) for n = 1..45</a>

%Y Cf. A111234, A308190, A308191, A308192.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jun 14 2019

%E a(17)-a(36) from _Chai Wah Wu_, Jun 14 2019

%E a(37)-a(38) from _Chai Wah Wu_, Jun 16 2019