A214780 - OEIS

A214780

a(1) = 2; a(n+1) = round(a(n)^(1 + 1/a(n))).

2, 3, 4, 6, 8, 10, 13, 16, 19, 22, 25, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 236, 242, 248, 254, 260, 266

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OFFSET

1,1

COMMENTS

Let b(1)= 2, b(n+1)= b(n)^(1 + 1/b(n)) for n > 0 and c(n) = round(b(n)). Let d(n) = a(n) - c(n). The d(n) sequence has a very interesting course: at the beginning of zero and the long cycles of positive and negative terms.

Firoozbakht conjecture: prime(n+1) < prime(n)^(1+1/n) for all n.

Let p(n+1) = p(n)^(1 + y(n)/p(n)). Conjecture: y(n) = O(log p(n)). The average value of y(n) is 1.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

FORMULA

a(n+1) = a(n) + log a(n) + O(1). a(n) ~ n log n. - Charles R Greathouse IV, Jul 30 2012

a(n) = n log n + n log log n + O(n) so a(n) = prime(n) + O(n). - Thomas Ordowski, May 07 2013

PROG

(PARI) a=[2]; for(n=2, 100, a=concat(a, round(a[n-1]^(1+1/a[n-1])))); a \\ Colin Barker, Jul 22 2014

CROSSREFS

Cf. A000040.

Sequence in context: A096061 A100919 A184109 * A024174 A008739 A280706

Adjacent sequences: A214777 A214778 A214779 * A214781 A214782 A214783

KEYWORD

nonn

AUTHOR

Thomas Ordowski, Jul 28 2012

STATUS

approved