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A343794 Numbers k > 0 such that 630*k + 315 is not an abundant number (A005101). 1
53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158, 165, 168, 173, 174, 176, 179, 183, 186, 189, 191, 194, 198, 200, 204, 209, 210, 215 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
630*k + 315 is an abundant number for the first 52 positive values of k.
The number of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 19, 276, 2242, 22249, 235300, 2319944, 22958712, 230566888, 2308563768, 23063629594, ... Apparently the asymptotic density of this sequence is 0.230...
There are 2048662 odd abundant numbers (A005231) below 10^9, of them 1213732 are of the form 630*k + 315. Apparently, the asymptotic density of abundant numbers of this form within the odd abundant numbers is about 0.6.
From Jianing Song, May 30 2022: (Start)
Numbers k > 0 such that (2*k+1)/sigma(2*k+1) <= 105/104.
Contains (p^i-1)/2 for all primes p >= 107 and i >= 1.
Since 315*p is abundant for primes p = 2, 3, 5, 7, 11, ..., 103, the prime factors of 2*k+1 are at least 107 if k is a term of this sequence. Hence we have a(n) = A005097(n+26) = (prime(n+27)-1)/2 for n <= 1354, whereas 2*a(1355)+1 = 11449 = 107^2.
The smallest term k such that 2*k+1 is not a prime power is k = a(4872), with 2*k+1 = 211*223. (End)
REFERENCES
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd ed., Penguin, 1997, p. 155.
M. T. Whalen and C. L. Miller, Odd abundant numbers: some interesting observations, Journal of Recreational Mathematics 22 (1990), pp. 257-261.
LINKS
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Vol. 37, No. 2 (2005), pp. 73-75.
EXAMPLE
53 is a term since 630*53 + 315 = 33705 is not an abundant number.
MATHEMATICA
abQ[n_] := DivisorSigma[1, n] > 2*n; Select[Range[200], !abQ[630*# + 315] &]
CROSSREFS
Sequence in context: A094462 A232125 A349091 * A042403 A217609 A239591
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 29 2021
STATUS
approved

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Last modified May 25 11:02 EDT 2024. Contains 372788 sequences. (Running on oeis4.)