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A275314 Euler's gradus ("suavitatis gradus", or degrees of softness) function.

%I A275314 #46 Jul 28 2022 09:07:57
%S A275314 1,2,3,3,5,4,7,4,5,6,11,5,13,8,7,5,17,6,19,7,9,12,23,6,9,14,7,9,29,8,
%T A275314 31,6,13,18,11,7,37,20,15,8,41,10,43,13,9,24,47,7,13,10,19,15,53,8,15,
%U A275314 10,21,30,59,9,61,32,11,7,17,14,67,19,25,12,71,8,73,38,11,21,17,16,79,9,9,42,83,11,21,44,31,14,89,10,19,25,33,48,23,8,97,14,15,11
%N A275314 Euler's gradus ("suavitatis gradus", or degrees of softness) function.
%C A275314 This sequence is described by Euler in the 1739 book "Tentamen", which provides numbers with gradus between 2 and 16 (page 41); the function is later used to calculate a measure of consonance of music intervals (e.g., see ratios on page 61). A description of Euler's function appears as a footnote in Helmholtz's "Sensations of Tone", which states that when p is prime, the degree of p is p; the degree of each other number is a product of prime numbers; and the degree of a product of two factors A and B, for which separately the numbers of degree are a and b respectively, is a + b - 1.
%H A275314 Michael S. Branicky, <a href="/A275314/b275314.txt">Table of n, a(n) for n = 1..10000</a>
%H A275314 David De Roure, P. Willcox, and D. M. Weigl, <a href="https://wp.nyu.edu/ismir2016/wp-content/uploads/sites/2294/2016/08/deroure-numbers.pdf">Numbers Into Notes: Cast Your Mind Back 200 Years</a>, Extended Abstracts for the Late-Breaking Demo Session of the 17th International Society for Music Information Retrieval Conference, 2016.
%H A275314 L. Euler, <a href="https://archive.org/stream/tentamennovaethe00eule#page/40/mode/2up">Tentamen novae theoriae mvsicae ex certissimis harmoniae principiis dilvcide expositae</a>, Petropoli, ex typographia Academiae scientiarvm, 1739, page 41.
%H A275314 Daniel Muzzulini, <a href="http://www.muzzulini.ch/publications/dm_1994_EulerOnConsonance.pdf">Leonhard Eulers Konsonanztheorie</a>, Musiktheorie 1994, 2, 135-146 (in German).
%H A275314 H. L. F. von Helmholtz, <a href="https://archive.org/stream/onsensationsofto00helmrich#page/230/mode/2up">On the sensations of tone as a physiological basis for the theory of music</a>, 1895, chapter XII. See Footnote p. 231.
%H A275314 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sensations_of_Tone">Sensations of Tone</a>.
%F A275314 If n = Product (p_j^k_j) then a(n) = 1 + Sum (k_j * (p_j - 1)).
%F A275314 a(n) = A001414(n) - A001222(n) + 1. - _Michel Marcus_, Jul 23 2016
%e A275314 For n = 5 the prime factors are 5 so a(5) = 1 + 4 = 5.
%e A275314 For n = 6 the prime factors are 2 and 3 so a(6) = 1 + (1 + 2) = 4.
%t A275314 Gradus[n_] := Plus @@ (Flatten[Table[#1, {#2}] & @@@ FactorInteger[n]] - 1) + 1
%o A275314 (PARI) a(n) = my(f = factor(n)); sum(k=1, #f~, (f[k,1]-1)*f[k,2])+ 1; \\ _Michel Marcus_, Jul 23 2016
%o A275314 (Python)
%o A275314 from sympy import factorint
%o A275314 def a(n): return 1 + sum(kj*(pj-1) for pj, kj in factorint(n).items())
%o A275314 print([a(n) for n in range(1, 101)]) # _Michael S. Branicky_, Dec 12 2021
%Y A275314 Cf. A001414, A001222.
%K A275314 nonn
%O A275314 1,2
%A A275314 _David De Roure_, Jul 22 2016