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

%I A381104 #27 Feb 14 2025 16:11:52
%S A381104 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,
%T A381104 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,
%U A381104 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
%N A381104 a(n) is the number of prime factors with exponent 1 in the prime factorization of the n-th superabundant number.
%C A381104 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.
%H A381104 Alois P. Heinz, <a href="/A381104/b381104.txt">Table of n, a(n) for n = 1..2000</a>
%H A381104 L. Alaoglu and P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1944-03.pdf">On highly composite and similar numbers,</a> Trans. Amer. Math. Soc., 56 (1944), 448-469. <a href="http://upforthecount.com/math/errata.html">Errata</a>.
%H A381104 Kevin Broughan, <a href="https://doi.org/10.1017/9781108178228">Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents</a>, Cambridge University Press, 2017.
%F A381104 a(n) = A056169(A004394(n)).
%e A381104 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.
%Y A381104 Cf. A004394, A056169.
%K A381104 nonn
%O A381104 1,4
%A A381104 _Agustin T. Besteiro_, Feb 14 2025