A343794 Numbers k > 0 such that 630*k + 315 is not an abundant number (A005101).

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

Offset: 1

Author

Amiram Eldar, Apr 29 2021

Keywords

• nonn

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)

Examples

			53 is a term since 630*53 + 315 = 33705 is not an abundant number.
		

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 113.
• Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Vol. 37, No. 2 (2005), pp. 73-75.

Crossrefs

Cf. A005101, A005231, A263837, A322287.

Programs

Mathematica

abQ[n_] := DivisorSigma[1, n] > 2*n; Select[Range[200], !abQ[630*# + 315] &]