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A004143
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From the powers that be.
(Formerly M1386)
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1
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0, 1, 2, 5, 10, 40, 40, 105, 5627, 14501, 330861, 658110, 897229, 26673531, 180566007, 180566007, 19299107624
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OFFSET
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1,3
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COMMENTS
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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
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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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R. B. Eggleton, P. Erdős and J. L. Selfridge, The powers that be, Amer. Math. Monthly, 83 (1976), 801-805.
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EXAMPLE
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a(8) = 105 from the chain of powers
2^105 > 3^66 > 5^45 > 7^37 > 11^30 > 13^28 > 17^25 > 19^24,
with each power satisfying p_i^{m_i} <= 2^105 < p_i^{m_i+1}. (End)
An independent calculation verifies these results:
2: 1 0
3: 2 1 0
4: 5 3 2 1
5: 10 6 4 3 2
6: 40 25 17 14 11 10
7: 40 25 17 14 11 10 9
8: 105 66 45 37 30 28 25 24
9: 5627 3550 2423 2004 1626 1520 1376 1324 1243
10: 14501 9149 6245 5165 4191 3918 3547 3413 3205 2984
11: 330861 208750 142494 117855 95640 89411 80945 77887 73141 68106 66783
12: 658110 415221 283432 234423 190236 177846 161006 154924 145484
135469 132838 126329
13: 897229 566088 386415 319599 259357 242465 219507 211215 198345
184691 181104 172230 167469 (End)
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CROSSREFS
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KEYWORD
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nonn,nice,more
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AUTHOR
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EXTENSIONS
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a(15) corrected, a(14) and a(16) confirmed, and a(17) from Bert Dobbelaere, Dec 26 2018
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STATUS
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approved
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