A381104 a(n) is the number of prime factors with exponent 1 in the prime factorization of the n-th superabundant number.

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

Offset

1,4

Comments

Alaoglu and Erdős proved that for all superabundant numbers, the exponents in their prime factorization are non-increasing. Moreover, there is always a sequence of prime factors with exponent 1 at the end of the factorization. The only exceptions for this sequence are 1, 4 and 36.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..2000

L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata.

Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017.

Formula

a(n) = A056169(A004394(n)).

Example

For n=8 the 8th superabundant number is 48 = 2^4*3^1. Only one prime factor appears with exponent 1 so a(8) = 1.

Crossrefs

Cf. A004394, A056169.

Sequence in context: A259538 A099314 A214341 * A281871 A131334 A004602

Adjacent sequences: A381100 A381101 A381102 * A381105 A381106 A381107

Keyword

nonn,new

Author

Agustin T. Besteiro, Feb 14 2025

Status

approved