A309042 Numbers k such that there is no prime p and index j > k such that A002182(j) = p * A002182(k).

2557, 6736, 9043, 9809, 13493, 15948, 16839, 20848, 23926, 29662, 30930, 31251, 31826, 33020, 35600, 36596, 54953, 56525, 59945, 59953, 64925, 66631, 69122, 69290, 70333, 70546, 77968, 78024, 83027, 84000, 84025, 91790, 91918, 100458, 100850, 101100, 107151

Offset

1,1

Comments

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

This is a sequence of counterexamples to the first conjecture by Alaoglu & Erdős that such highly composite numbers do not exist. Robin found the first counterexample: A002182(2557) = 3.000... * 10^153. The sequence of counterexamples to their second conjecture is A272606.

Links

Amiram Eldar, Table of n, a(n) for n = 1..122 (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, A272606.

Sequence in context: A256948 A202587 A125492 * A170794 A068265 A252677

Adjacent sequences: A309039 A309040 A309041 * A309043 A309044 A309045

Keyword

nonn

Author

Amiram Eldar, Jul 09 2019

Status

approved