A004143 From the powers that be. (Formerly M1386)

0, 1, 2, 5, 10, 40, 40, 105, 5627, 14501, 330861, 658110, 897229, 26673531, 180566007, 180566007, 19299107624

Offset

1,3

Comments

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

References

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

Links

Table of n, a(n) for n=1..17.

R. B. Eggleton, P. Erdős and J. L. Selfridge, The powers that be, Amer. Math. Monthly, 83 (1976), 801-805.

Example

From Sean A. Irvine, Dec 22 2015: (Start)
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)
From Don Reble, Dec 22 2015: (Start)
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)

Crossrefs

Cf. A266162 (erroneous version).

Sequence in context: A363429 A290032 A155217 * A266162 A103036 A018439

Adjacent sequences: A004140 A004141 A004142 * A004144 A004145 A004146

Keyword

nonn,nice,more

Author

N. J. A. Sloane

Extensions

Offset, a(8) corrected and a(13) from Sean A. Irvine, Dec 22 2015

a(14)-a(16) from Charlie Neder, Nov 04 2018

a(15) corrected, a(14) and a(16) confirmed, and a(17) from Bert Dobbelaere, Dec 26 2018

Status

approved