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| In [[mathematics]], a '''sparsely totient number''' is a certain kind of [[natural number]]. A natural number, ''n'', is sparsely totient if for all ''m'' > ''n'',
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| :φ(''m'')>φ(''n''),
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| where φ is [[Euler's totient function]]. The first few sparsely totient numbers are:
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| [[2 (number)|2]], [[6 (number)|6]], [[12 (number)|12]], [[18 (number)|18]], [[30 (number)|30]], [[42 (number)|42]], [[60 (number)|60]], [[66 (number)|66]], [[90 (number)|90]], [[120 (number)|120]], [[126 (number)|126]], [[150 (number)|150]], [[210 (number)|210]], [[240 (number)|240]], [[270 (number)|270]], [[330 (number)|330]], [[420 (number)|420]], 462, 510, 630 {{OEIS|id=A036913}}.
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| For example, 18 is a sparsely totient number because φ(18) = 6, and any number ''m'' > 18 falls into at least one of the following classes:
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| #''m'' has a prime factor ''p'' ≥ 11, so φ(''m'') ≥ φ(11) = 10 > φ(18).
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| #''m'' is a multiple of 7 and ''m''/7 ≥ 3, so φ(''m'') ≥ 2φ(7) = 12 > φ(18).
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| #''m'' is a multiple of 5 and ''m''/5 ≥ 4, so φ(''m'') ≥ 2φ(5) = 8 > φ(18).
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| #''m'' is a multiple of 3 and ''m''/3 ≥ 7, so φ(''m'') ≥ 4φ(3) = 8 > φ(18).
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| #''m'' is a power of 2 and ''m'' ≥ 32, so φ(''m'') ≥ φ(32) = 16 > φ(18).
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| The concept was introduced by [[David Masser]] and [[Peter Shiu]] in 1986.
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| ==Properties==
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| * If ''P''(''n'') is the largest [[prime factor]] of ''n'', then <math>\liminf P(n)/\log n=1</math>.
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| * <math>P(n)\ll \log^\delta n</math> holds for an exponent <math>\delta=37/20</math>.
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| * It is conjectured that <math>\limsup P(n) / \log n = 2</math>.
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| ==References==
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| {{reflist}}
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| * {{cite journal | last1=Baker | first1=Roger C. | last2=Harman | first2=Glyn | author2-link=Glyn Harman | title=Sparsely totient numbers | journal=Ann. Fac. Sci. Toulouse, VI. Sér., Math. | volume=5 | number=2 | pages=183-190 | year=1996 | issn=0240-2963 | zbl=0871.11060 | url=https://eudml.org/doc/73381 }}
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| * {{cite journal | zbl=0538.10006 | last1=Masser | first1=D.W. | author1-link=David Masser | last2=Shiu | first2=P. | title=On sparsely totient numbers | journal=Pac. J. Math. | volume=121 | pages=407-426 | year=1986 | issn=0030-8730 | zbl=0538.10006 | url=http://projecteuclid.org/euclid.pjm/1102702441 | mr=819198 }}
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| {{Totient}}
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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