sparsely totient number

A sparsely totient number is the largest integer with a given totient. That is, given an sparsely totient number $n$ , the inequality $\\phi(m)>\\phi(n)$ (with $\\phi(x)$ being Euler’s totient function) is true for any $m>n$ . For example, $\\phi(12)=4$ . We can then verify that $\\phi(13)=12$ , $\\phi(14)=6$ and $\\phi(15)=8$ . Accepting as true that the inequality $\\phi(n)>\\sqrt{n}$ holds for all $n>6$ , we don’t need to check any larger numbers to confirm that 12 is the largest integer with 4 as its totient.

The first few sparsely totient numbers are 2, 6, 12, 18, 30, 42, 60, 66, 90, 120, 126, 150, 210, 240, 270, 330, 420, 462, 510, 630, 660, 690, 840, 870, etc., listed in A036913 of Sloane’s OEIS.

All sparsely totient numbers are even. In 1986, Masser and Shiu proved that the $i$ th primorial multiplied by the $i$ th prime (for $i>1$ , thus: 18, 150, 1470, 25410, 390390, etc.) is a sparsely totient number.

References

1 Roger C. Baker & Glyn Harman, “Sparsely totient numbers,” Annales de la faculte des sciences de Toulouse Ser. 6 5 no. 2 (1996): 183 - 190
2 D. W. Masser & P. Shiu, “On sparsely totient numbers,” Pacific J. Math. 121, no. 2 (1986): 407 - 426.

单词	SparselyTotientNumber
释义	sparsely totient number A sparsely totient number is the largest integer with a given totient. That is, given an sparsely totient number $n$ , the inequality $\\phi(m)>\\phi(n)$ (with $\\phi(x)$ being Euler’s totient function) is true for any $m>n$ . For example, $\\phi(12)=4$ . We can then verify that $\\phi(13)=12$ , $\\phi(14)=6$ and $\\phi(15)=8$ . Accepting as true that the inequality $\\phi(n)>\\sqrt{n}$ holds for all $n>6$ , we don’t need to check any larger numbers to confirm that 12 is the largest integer with 4 as its totient. The first few sparsely totient numbers are 2, 6, 12, 18, 30, 42, 60, 66, 90, 120, 126, 150, 210, 240, 270, 330, 420, 462, 510, 630, 660, 690, 840, 870, etc., listed in A036913 of Sloane’s OEIS. All sparsely totient numbers are even. In 1986, Masser and Shiu proved that the $i$ th primorial multiplied by the $i$ th prime (for $i>1$ , thus: 18, 150, 1470, 25410, 390390, etc.) is a sparsely totient number. References 1 Roger C. Baker & Glyn Harman, “Sparsely totient numbers,” Annales de la faculte des sciences de Toulouse Ser. 6 5 no. 2 (1996): 183 - 190 2 D. W. Masser & P. Shiu, “On sparsely totient numbers,” Pacific J. Math. 121, no. 2 (1986): 407 - 426.
随便看	Surjective surjective SurjectiveHomomorphismBetweenUnitaryRings surjective homomorphism between unitary rings SurjectiveOpenMapsInTermsOfNets AbelianGroupIsDivisibleIfAndOnlyIfItIsAnInjectiveObject abelian group is divisible if and only if it is an injective object AbelianGroupsOfOrder120 abelian groups of order 120 Abelianization abelianization AbelianNumberField abelian number field AbelianVariety abelian variety AbelPrize Abel Prize AbelsLemma AbelsLimitTheorem AbelsMultiplicationRuleForSeries AbelSummability Abel summability Abel’s lemma Abel’s limit theorem Abel’s multiplication rule for series