A272606 Numbers k such that there is no prime p and index j < k such that A002182(k) = p * A002182(j).

1, 815, 4372, 4996, 5312, 5442, 22093, 24931, 24964, 25587, 26064, 28776, 29365, 29372, 32757, 34115, 34122, 36046, 51207, 52527, 54746, 57927, 58971, 63160, 63988, 63993, 82127, 95661, 95746, 97931, 128049, 132331, 132720, 134358, 136254, 150282, 179341

Offset

1,2

Comments

Indices k such that the k-th highly composite number cannot be obtained by multiplying any smaller highly composite number by a prime.

This is a sequence of counterexamples to the second conjecture by Alaoglu & Erdős that such highly composite numbers do not exist (they did not consider 1 to be highly composite number). Robin found the first 3 counterexamples: A002182(815) = 3.622... * 10^65, A002182(4372) = 6.043... * 10^220, and A002182(4996) = 1.115 * 10^244. - Amiram Eldar, Jul 09 2019

Links

Amiram Eldar, Table of n, a(n) for n = 1..77 (Calculated from Achim Flammenkamp's List of the first 779,674 highly composite numbers)

Leonidas Alaoglu and Paul Erdős, On highly composite and similar numbers, Transactions of the American Mathematical Society, Vol. 56, No. 3 (1944), pp. 448-469. See p. 467.

Achim Flammenkamp, Highly Composite Numbers.

Guy Robin, Méthodes d'optimisation pour un problème de théorie des nombres, RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Vol. 17, No. 3 (1983), pp. 239-247.

Crossrefs

Cf. A002182.

Sequence in context: A035854 A099116 A183820 * A235976 A015159 A250803

Adjacent sequences: A272603 A272604 A272605 * A272607 A272608 A272609

Keyword

nonn

Author

Joerg Arndt, Nov 01 2016

Extensions

a(7)-a(37) from Amiram Eldar, Jul 09 2019

Status

approved