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From the powers that be.
(Formerly M1386)
1

%I M1386 #48 Dec 27 2018 03:17:11

%S 0,1,2,5,10,40,40,105,5627,14501,330861,658110,897229,26673531,

%T 180566007,180566007,19299107624

%N From the powers that be.

%C For n>0, let b be the smallest nonnegative integer such that 2^m_1 > 3^m_2 > ... > prime(n)^m_n, where m_i is the exponent satisfying prime(i)^m_i <= b < prime(i)^(m_i+1). This sequence records the exponent m_1 for b because b=2^m_1. - _Tom Edgar_, Dec 05 2014

%C Equivalently, a(n) is the first k such that p^frac(k/log_2(p)) is increasing over the first n primes. - _Charlie Neder_, Nov 03 2018

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H R. B. Eggleton, P. Erdős and J. L. Selfridge, <a href="http://www.jstor.org/stable/2318687">The powers that be</a>, Amer. Math. Monthly, 83 (1976), 801-805.

%e From _Sean A. Irvine_, Dec 22 2015: (Start)

%e a(8) = 105 from the chain of powers

%e 2^105 > 3^66 > 5^45 > 7^37 > 11^30 > 13^28 > 17^25 > 19^24,

%e with each power satisfying p_i^{m_i} <= 2^105 < p_i^{m_i+1}. (End)

%e From _Don Reble_, Dec 22 2015: (Start)

%e An independent calculation verifies these results:

%e 2: 1 0

%e 3: 2 1 0

%e 4: 5 3 2 1

%e 5: 10 6 4 3 2

%e 6: 40 25 17 14 11 10

%e 7: 40 25 17 14 11 10 9

%e 8: 105 66 45 37 30 28 25 24

%e 9: 5627 3550 2423 2004 1626 1520 1376 1324 1243

%e 10: 14501 9149 6245 5165 4191 3918 3547 3413 3205 2984

%e 11: 330861 208750 142494 117855 95640 89411 80945 77887 73141 68106 66783

%e 12: 658110 415221 283432 234423 190236 177846 161006 154924 145484

%e 135469 132838 126329

%e 13: 897229 566088 386415 319599 259357 242465 219507 211215 198345

%e 184691 181104 172230 167469 (End)

%Y Cf. A266162 (erroneous version).

%K nonn,nice,more

%O 1,3

%A _N. J. A. Sloane_

%E Offset, a(8) corrected and a(13) from _Sean A. Irvine_, Dec 22 2015

%E a(14)-a(16) from _Charlie Neder_, Nov 04 2018

%E a(15) corrected, a(14) and a(16) confirmed, and a(17) from _Bert Dobbelaere_, Dec 26 2018