A217660 -id:A217660 - cp's OEIS frontend

A280235 Constant appearing in the Nicolas-Robin bound for the divisor function.

1, 5, 3, 7, 9, 3, 9, 8, 6, 0, 6, 7, 5, 1, 2, 6, 1, 7, 4, 9, 5, 7, 9, 0, 8, 6, 0, 7, 3, 1, 2, 1, 2, 2, 1, 3, 6, 7, 4, 9, 8, 6, 3, 1, 0, 8, 4, 2, 5, 2, 1, 0, 7, 6, 2, 2, 1, 4, 5, 7, 2, 3, 5, 7, 9, 4, 3, 1, 1, 9, 6, 6, 9, 3, 3, 8, 3, 5, 1, 4, 1, 7, 0, 5, 4, 4, 7, 9, 3

Offset: 1

Charles R Greathouse IV, Feb 25 2017

The number of divisors of n is at most 2^(k * log n/log log n) where k is this constant. Equality is attained precisely at n = 6983776800.

			1.53793986067512617495790860731212213674986310842521076221457235794311...

Indranil Ghosh, Table of n, a(n) for n = 1..150
J. L. Nicolas and G. Robin, Majorations explicites pour le nombre de diviseurs de N, Canadian Mathematical Bulletin 26 (1983), pp. 485-492.

Cf. A217660.

L = Log[6983776800]; RealDigits[2 * Log[48] * Log[L] / L / Log[2], 10, 89][[1]] (* Indranil Ghosh, Mar 12 2017 *)

L=log(6983776800); 2*log(48)*log(L)/L/log(2)

A309966 Numbers n > 1 that give record values for f(n) = sigma(n)/(nlog(log(3d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).

Original entry on oeis.org

2, 116288545977326780410953600, 581442729886633902054768000, 7093601304616933605068169600, 35468006523084668025340848000, 475271287409334551539567363200, 2376356437046672757697836816000, 168721307030313765796546413936000, 1855934377333451423762010553296000

Offset: 1

Amiram Eldar, Aug 25 2019

Nicolas proved that f(n) reaches its maximum at n = 2^3 * (3#)^2 * 5# * 13# * 113# = 8201519488959040182625924708238885435575055666675808000 ~ 8.2 * 10^54 which is the last term of this sequence (prime(n)# = A002110(n) is the n-th primorial).

Amiram Eldar, Table of n, a(n) for n = 1..26
Jean-Louis Nicolas, Quelques inégalités effectives entre des fonctions arithmétiques usuelles, Functiones et Approximatio, Vol. 39, No. 2 (2008), pp. 315-334. See Theorem 1.2.

Cf. A000005, A000203, A217660.

Subsequence of A025487.

A309968 Numbers n > 1 that give record values for f(n) = sigma(n)/n - e^gamma * log(log(ed(n))) - e^gamma log(log(log(e^e * d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).

Original entry on oeis.org

2, 74801040398884800, 224403121196654400, 3066842656354276800, 6133685312708553600, 9200527969062830400, 18401055938125660800, 131874234223233902400, 263748468446467804800, 395622702669701707200, 791245405339403414400, 6198089008491993412800, 12396178016983986825600

Offset: 1

Amiram Eldar, Aug 25 2019

Nicolas proved that f(n) reaches its maximum at n = 2^7 * (3#)^4 * 5# * (7#)^2 * 19# * 47# * 277# * 45439# ~ 8.0244105... * 10^19786 which is the last term of this sequence (prime(n)# = A002110(n) is the n-th primorial).

Amiram Eldar, Table of n, a(n) for n = 1..68
Jean-Louis Nicolas, Quelques inégalités effectives entre des fonctions arithmétiques usuelles, Functiones et Approximatio, Vol. 39, No. 2 (2008), pp. 315-334. See Theorem 1.2.

Cf. A000005, A000203, A073004, A073226, A217660.

Subsequence of A025487.

cp's OEIS Frontend

A280235 Constant appearing in the Nicolas-Robin bound for the divisor function.

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A309966 Numbers n > 1 that give record values for f(n) = sigma(n)/(nlog(log(3d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).

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A309968 Numbers n > 1 that give record values for f(n) = sigma(n)/n - e^gamma * log(log(ed(n))) - e^gamma log(log(log(e^e * d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).

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cp's OEIS Frontend

A280235 Constant appearing in the Nicolas-Robin bound for the divisor function.

Views

Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

A309966 Numbers n > 1 that give record values for f(n) = sigma(n)/(n*log(log(3*d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).

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A309968 Numbers n > 1 that give record values for f(n) = sigma(n)/n - e^gamma * log(log(e*d(n))) - e^gamma * log(log(log(e^e * d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).

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A309966 Numbers n > 1 that give record values for f(n) = sigma(n)/(nlog(log(3d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).

A309968 Numbers n > 1 that give record values for f(n) = sigma(n)/n - e^gamma * log(log(ed(n))) - e^gamma log(log(log(e^e * d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).