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A275314
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Euler's gradus ("suavitatis gradus", or degrees of softness) function.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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If n = Product (p_j^k_j) then a(n) = 1 + Sum (k_j * (p_j - 1)).
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EXAMPLE
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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.
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MATHEMATICA
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Gradus[n_] := Plus @@ (Flatten[Table[#1, {#2}] & @@@ FactorInteger[n]] - 1) + 1
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PROG
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(PARI) a(n) = my(f = factor(n)); sum(k=1, #f~, (f[k, 1]-1)*f[k, 2])+ 1; \\ Michel Marcus, Jul 23 2016
(Python)
from sympy import factorint
def a(n): return 1 + sum(kj*(pj-1) for pj, kj in factorint(n).items())
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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