A275314 Euler's gradus ("suavitatis gradus", or degrees of softness) function.
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, 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, 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
Offset: 1
Keywords
Examples
For n = 5 the prime factors are 5 so a(5) = 1 + 4 = 5. For n = 6 the prime factors are 2 and 3 so a(6) = 1 + (1 + 2) = 4.
Links
- Michael S. Branicky, Table of n, a(n) for n = 1..10000
- David De Roure, P. Willcox, and D. M. Weigl, Numbers Into Notes: Cast Your Mind Back 200 Years, Extended Abstracts for the Late-Breaking Demo Session of the 17th International Society for Music Information Retrieval Conference, 2016.
- L. Euler, Tentamen novae theoriae mvsicae ex certissimis harmoniae principiis dilvcide expositae, Petropoli, ex typographia Academiae scientiarvm, 1739, page 41.
- Daniel Muzzulini, Leonhard Eulers Konsonanztheorie, Musiktheorie 1994, 2, 135-146 (in German).
- H. L. F. von Helmholtz, On the sensations of tone as a physiological basis for the theory of music, 1895, chapter XII. See Footnote p. 231.
- Wikipedia, Sensations of Tone.
Programs
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Mathematica
Gradus[n_] := Plus @@ (Flatten[Table[#1, {#2}] & @@@ FactorInteger[n]] - 1) + 1 -
PARI
a(n) = my(f = factor(n)); sum(k=1, #f~, (f[k,1]-1)*f[k,2])+ 1; \\ Michel Marcus, Jul 23 2016
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Python
from sympy import factorint def a(n): return 1 + sum(kj*(pj-1) for pj, kj in factorint(n).items()) print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Dec 12 2021
Formula
If n = Product (p_j^k_j) then a(n) = 1 + Sum (k_j * (p_j - 1)).
Comments