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

Author

Agustin T. Besteiro, Feb 14 2025

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A004394, A056169.

Formula

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