The OEIS is supported by the many generous donors to the OEIS Foundation.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #21 Jan 08 2025 11:23:20
%S 53,54,56,63,65,68,69,74,75,78,81,83,86,89,90,95,96,98,99,105,111,113,
%T 114,116,119,120,125,128,131,134,135,138,140,141,146,153,155,156,158,
%U 165,168,173,174,176,179,183,186,189,191,194,198,200,204,209,210,215
%N Numbers k > 0 such that 630*k + 315 is not an abundant number (A005101).
%C 630*k + 315 is an abundant number for the first 52 positive values of k.
%C 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...
%C 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.
%C From _Jianing Song_, May 30 2022: (Start)
%C Numbers k > 0 such that (2*k+1)/sigma(2*k+1) <= 105/104.
%C Contains (p^i-1)/2 for all primes p >= 107 and i >= 1.
%C 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.
%C 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)
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd ed., Penguin, 1997, p. 155.
%D M. T. Whalen and C. L. Miller, Odd abundant numbers: some interesting observations, Journal of Recreational Mathematics 22 (1990), pp. 257-261.
%H Amiram Eldar, <a href="/A343794/b343794.txt">Table of n, a(n) for n = 1..10000</a>
%H Jean-Marie De Koninck, <a href="https://bookstore.ams.org/mbk-64/133">Those Fascinating Numbers</a>, Amer. Math. Soc., (2009), page 113.
%H Jay L. Schiffman, <a href="http://www.appliedprobability.org/data/files/MS%20issues/Vol37_No2.pdf">Odd Abundant Numbers</a>, Mathematical Spectrum, Vol. 37, No. 2 (2005), pp. 73-75.
%e 53 is a term since 630*53 + 315 = 33705 is not an abundant number.
%t abQ[n_] := DivisorSigma[1, n] > 2*n; Select[Range[200], !abQ[630*# + 315] &]
%Y Cf. A005101, A005231, A263837, A322287.
%K nonn,changed
%O 1,1
%A _Amiram Eldar_, Apr 29 2021