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A353537
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Numbers whose abundancy index is larger than Pi^2/6.
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7
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4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 148, 150
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OFFSET
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1,1
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COMMENTS
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The abundancy index of a number k is sigma(k)/k, where sigma is the sum of divisors function (A000203).
Pi^2/6 (A013661) is the asymptotic mean of the abundancy indices of the positive integers.
The least odd term is 45 and the least term that is coprime to 6 is 25025.
Davenport (1933) proved that sigma(k)/k possesses a continuous distribution function and that the asymptotic density of numbers with abundancy index that is larger than x exists for all x > 1 and is a continuous function of x. Therefore, this sequence has an asymptotic density.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 4, 41, 436, 4258, 42928, 428557, 4286145, 42864566, 428585795, 4286368677, 42861854540, ... Apparently, the asymptotic density is 0.4286... which means that the distribution of the abundancy indices is skewed with a positive nonparametric skew.
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REFERENCES
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Harold Davenport, Über numeri abundantes, Sitzungsberichte der Preußischen Akademie der Wissenschaften, phys.-math. Klasse, No. 6 (1933), pp. 830-837.
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LINKS
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EXAMPLE
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4 is a term since sigma(4)/4 = 7/4 = 1.75 > Pi^2/6 = 1.644...
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MATHEMATICA
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Select[Range[150], DivisorSigma[-1, #] > Pi^2/6 &]
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PROG
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(PARI) isok(k) = sigma(k)/k > Pi^2/6; \\ Michel Marcus, Apr 25 2022
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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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