

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
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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.
Numbers k > 0 such that (2*k+1)/sigma(2*k+1) <= 105/104.
Contains (p^i1)/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. 257261.


LINKS



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



KEYWORD

nonn


AUTHOR



STATUS

approved



