A308193 Indices of records in A308190.

5, 6, 7, 10, 16, 17, 29, 53, 101, 197, 389, 773, 1542, 3079, 6154, 12304, 24604, 36901, 73798, 147592, 295180, 295517, 591030, 1182056, 1574849, 3149694, 4728211, 6299383, 12598762, 25197520, 25197533, 50395062, 100790119, 201580234, 403160464, 806320924, 1232145821, 2464291638

Offset: 1

Author

N. J. A. Sloane, Jun 14 2019

Keywords

• nonn

Comments

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?

From Chai Wah Wu, Jun 14 2019: (Start)

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

Theorem:

1. All terms are even or prime.

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

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

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.

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.

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

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.

(End)

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..45

Crossrefs

Cf. A111234, A308190, A308191, A308192.

Extensions

a(17)-a(36) from Chai Wah Wu, Jun 14 2019

a(37)-a(38) from Chai Wah Wu, Jun 16 2019