Proofs
The proofs of the theorems are long; perhaps they should be shortened?
For example, Theorem 1 follows almost immediately from $\sigma _{-1}(n)\geq 1$ and Theorem 2 follows from Sylvester's theorem on the Frobenius problem (with 18 and 20), plus finite checking on 48-142.
Charles R Greathouse IV 17:04, 29 July 2012 (UTC)
- Perhaps. But before shortening Theorem 2, I'd like for there to be a page about Sylvester's theorem with at least a placeholder for the proof. Alonso del Arte 23:35, 29 July 2012 (UTC)
Properties
Density of the odd abundant numbers
For even abundant numbers, we have
Abundant numbers (and hence even abundant numbers) are closed under multiplication by arbitrary positive integers: any positive multiple of an abundant number is abundant.
As a consequence, the even abundant numbers are of positive density (among the positive integers). In particular, their lower density (among the positive integers) is at least 0.2453 and their upper density is at most 0.2460.
Are the odd abundant numbers of positive density?
For odd abundant numbers, we have
(1) The first odd abundant number is the 232th abundant number!
(2) Odd abundant numbers are closed under multiplication by arbitrary positive odd integers: any positive odd multiple of an odd abundant number is odd and abundant.
Now (1) seems to suggest that the odd abundant numbers have a much lower density, if not 0, than the even abundant numbers.
But (2) combined with the fact that half the positive integers are odd, and if (1) happens to be a ground effect, seems to suggest that the density of odd abundant numbers might be half the density of even abundant numbers, does that make sense? — Daniel Forgues 07:09, 30 July 2012 (UTC)
- Since all multiples of even abundant numbers are even abundant numbers, while only the odd multiples of odd abundant numbers are odd abundant numbers, I would expect that there are asymptotically twice as many even abundant numbers as there are odd abundant numbers, thus a density (among the abundant numbers) of 2/3 for the even abundant numbers and 1/3 for the odd abundant numbers. — Daniel Forgues 03:01, 24 September 2016 (UTC)
- Not quite! All even and odd multiples of even abundant numbers are even abundant numbers, but also all even multiples of odd abundant numbers are even abundant numbers, while only all odd multiples of odd abundant numbers are odd abundant numbers. I would then expect that there are asymptotically thrice as many even abundant numbers as there are odd abundant numbers, thus a density (among the abundant numbers) of 3/4 for the even abundant numbers and 1/4 for the odd abundant numbers. — Daniel Forgues 03:07, 24 September 2016 (UTC)
- Yes, there are only 90 odd abundant numbers less than 40000, while there are 9822 even abundant numbers less than 40000, but I believe this may be a ground effect due to the fact that the first odd abundant number is 945 (thus limiting the number of odd multiples). — Daniel Forgues 03:19, 24 September 2016 (UTC)
- The above reasoning is probably valid only if most of the abundant numbers are nonprimitive abundant numbers (abundant numbers which are a multiple of an abundant number). — Daniel Forgues 03:39, 24 September 2016 (UTC)
- The density of odd abundant numbers is positive. (All odd multiples of 945 are abundant, so their density is at least 1/1890.) More generally, for any x and y, the density of numbers coprime to y# with abundancy greater than x is positive. This follows from the divergence of the prime harmonic: essentially you start from y+1 and take primes until the sum of their inverses is larger than log x, which will be about exp(exp(log x + log log y)). This is a first-order approximation of the product (1 + 1/p) you get for sigma_-1.
- I expect the density of odd abundant numbers in the abundant numbers is much smaller than 1/4. Probably between 1/100 and 1/150, if I had to guess. I think some calculations have been done... by Iannucci, maybe?
- Charles R Greathouse IV 07:26, 24 September 2016 (UTC)
- Do you know whether or not it be the case that asymptotically 100% of the abundant numbers are nonprimitive abundant numbers? If that be the case, would it imply that asymptotically 1/4 of the abundant numbers would be odd? — Daniel Forgues 23:05, 24 September 2016 (UTC)
- I'm hinting that asymptotically 100% of abundant numbers are nonprimitive, but I suspect that the ratio of odd abundant numbers to abundant numbers might be equal to 1/4 of the ratio of primitive odd abundant numbers to primitive abundant numbers (where primitive would stand for not a multiple of either a perfect number or an abundant number). — Daniel Forgues 23:57, 25 September 2016 (UTC)
Sum of two abundant numbers
If you wanted a proof that all large enough numbers are the sum of two abundant numbers, you could use 945 plus one of these even abundant numbers:
1890, 21736, 43472, 948, 3784, 2840, 1896, 952, 7568, 954, 1900, 4736, 12, 33088, 1904, 960, 15136, 29312, 18, 19864, 20, 966, 9472, 968, 24, 2860, 5696, 972, 3808, 40664, 30, 8536, 7592, 978, 5704, 980, 36, 14212, 35948, 984, 40, 6656, 42, 19888, 18944, 990, 1936, 992, 48, 2884, 1940, 996, 11392, 25568, 54, 1000, 56, 1002, 13288, 4784, 60, 33136, 1952, 1008, 15184, 2900, 66, 27472, 3848, 1014, 70, 8576, 72, 6688, 20864, 1020, 30316, 2912, 78, 46384, 80, 1026, 13312, 35048, 84, 2920, 1976, 1032, 88, 21824, 90, 1036, 3872, 1038, 1984, 1040, 96, 29392, 1988, 1044, 100, 10496, 102, 69088, 104, 1050, 5776, 19952, 108, 2944, 2000, 1056, 112, 14288, 114, 1060, 26576, 1062, 9568, 1064, 120, 25636, 51152, 1068, 3904, 2960, 126, 8632, 30368, 1074, 2020, 12416, 132, 2968, 2024, 1080, 7696, 29432, 138, 4864, 140, 1086, 9592, 1088, 144, 2980, 13376, 1092, 41728, 71024, 150, 38896, 11492, 1098, 2044, 1100, 156, 2992, 17168, 1104, 160, 2996, 162, 4888, 3944, 1110, 32296, 38912, 168, 6784, 2060, 1116, 3952, 3008, 174, 1120, 176, 1122, 5848, 12464, 180, 3016, 2072, 1128, 7744, 3020, 186, 31372, 3968, 1134, 2080, 16256, 192, 6808, 17204, 1140, 196, 40832, 198, 1144, 200, 1146, 20992, 1148, 204, 3040, 9656, 1152, 208, 14384, 210, 12496, 11552, 1158, 32344, 1160, 216, 3052, 5888, 1164, 220, 18176, 222, 23848, 224, 1170, 5896, 12512, 228, 10624, 2120, 1176, 22912, 6848, 234, 1180, 19136, 1182, 2128, 1184, 240, 14416, 21032, 1188, 64504, 1190, 246, 46552, 7808, 1194, 2140, 20096, 252, 10648, 17264, 1200, 15376, 14432, 258, 1204, 260, 1206, 17272, 16328, 264, 3100, 2156, 1212, 4048, 18224, 270, 1216, 272, 1218, 9724, 1220, 276, 10672, 9728, 1224, 280, 14456, 282, 20128, 19184, 1230, 2176, 1232, 288, 14464, 2180, 1236, 26752, 3128, 294, 1240, 30536, 1242, 36208, 12584, 300, 3136, 28652, 1248, 304, 3140, 306, 5032, 308, 1254, 2200, 8816, 312, 22048, 5984, 1260, 34336, 10712, 318, 31504, 320, 1266, 2212, 35288, 324, 3160, 9776, 1272, 7888, 3164, 330, 5056, 34352, 1278, 24904, 1280, 336, 6952, 13568, 1284, 340, 10736, 342, 1288, 7904, 1290, 6016, 50432, 348, 10744, 350, 1296, 352, 6968, 354, 1300, 4136, 1302, 17368, 39104, 360, 6976, 6032, 1308, 364, 3200, 366, 1312, 368, 1314, 2260, 1316, 372, 41008, 21164, 1320, 7936, 6992, 378, 5104, 380, 1326, 32512, 8888, 384, 1330, 13616, 1332, 30628, 3224, 390, 16456, 392, 1338, 13624, 1340, 396, 10792, 2288, 1344, 400, 25916, 402, 12688, 26864, 1350, 2296, 1352, 408, 22144, 2300, 1356, 11752, 3248, 414, 1360, 416, 1362, 21208, 27824, 420, 3256, 24992, 1368, 23104, 3260, 426, 1372, 19328, 1374, 2320, 1376, 432, 14608, 2324, 1380, 4216, 18392, 438, 8944, 440, 1386, 36352, 5168, 444, 3280, 47696, 1392, 448, 29744, 450, 42976, 11792, 1398, 25024, 1400, 456, 7072, 21248, 1404, 460, 33536, 462, 1408, 464, 1410, 6136, 5192, 468, 3304, 2360, 1416, 30712, 10868, 474, 1420, 476, 1422, 2368, 16544, 480, 37336, 28832, 1428, 4264, 1430, 486, 20332, 23168, 1434, 490, 20336, 492, 3328, 9944, 1440, 8056, 3332, 498, 12784, 500, 1446, 2392, 24128, 504, 3340, 13736, 1452, 4288, 3344, 510, 1456, 23192, 1458, 28864, 1460, 516, 10912, 2408, 1464, 520, 14696, 522, 5248, 34544, 1470, 17536, 1472, 528, 7144, 2420, 1476, 532, 26048, 534, 1480, 8096, 1482, 6208, 1484, 540, 18496, 2432, 1488, 544, 3380, 546, 39292, 19448, 1494, 550, 1496, 552, 3388, 21344, 1500, 19456, 3392, 558, 1504, 560, 1506, 6232, 12848, 564, 3400, 17576, 1512, 38368, 22304, 570, 20416, 572, 1518, 2464, 1520, 576, 7192, 6248, 1524, 580, 3416, 582, 9088, 15704, 1530, 6256, 5312, 588, 18544, 2480, 1536, 30832, 7208, 594, 1540, 27056, 1542, 25168, 46904, 600, 7216, 2492, 1548, 19504, 3440, 606, 9112, 608, 1554, 2500, 5336, 612, 11008, 10064, 1560, 616, 7232, 618, 24244, 620, 1566, 6292, 1568, 624, 3460, 44096, 1572, 4408, 11024, 630, 39376, 15752, 1578, 21424, 1580, 636, 3472, 10088, 1584, 640, 45056, 642, 5368, 644, 1590, 25216, 9152, 648, 14824, 650, 1596, 19552, 33728, 654, 1600, 8216, 1602, 2548, 20504, 660, 3496, 2552, 1608, 34684, 1610, 666, 28072, 8228, 1614, 2560, 31856, 672, 14848, 6344, 1620, 27136, 71552, 678, 1624, 680, 1626, 21472, 5408, 684, 3520, 2576, 1632, 15808, 7304, 690, 9196, 4472, 1638, 2584, 1640, 696, 14872, 21488, 1644, 700, 3536, 702, 16768, 704, 1650, 13936, 1652, 708, 26224, 2600, 1656, 8272, 22448, 714, 1660, 12056, 1662, 55528, 1664, 720, 3556, 6392, 1668, 12064, 3560, 726, 1672, 728, 1674, 2620, 5456, 732, 11128, 2624, 1680, 736, 11132, 738, 13024, 740, 1686, 2632, 9248, 744, 3580, 36656, 1692, 748, 3584, 750, 1696, 15872, 1698, 6424, 1700, 756, 11152, 10208, 1704, 760, 22496, 762, 1708, 4544, 1710, 17776, 9272, 768, 7384, 770, 1716, 23452, 3608, 774, 1720, 61256, 1722, 6448, 5504, 780, 11176, 17792, 1728, 784, 3620, 786, 5512, 27248, 1734, 2680, 1736, 792, 22528, 6464, 1740, 4576, 26312, 798, 16864, 800, 1746, 36712, 28208, 804, 1750, 17816, 1752, 15928, 7424, 810, 35776, 812, 1758, 2704, 1760, 816, 45232, 40508, 1764, 820, 7436, 822, 1768, 23504, 1770, 2716, 13112, 828, 33904, 2720, 1776, 832, 3668, 834, 1780, 836, 1782, 2728, 9344, 840, 22576, 6512, 1788, 4624, 3680, 846, 1792, 31088, 1794, 2740, 5576, 852, 11248, 2744, 1800, 31096, 41492, 858, 43384, 860, 1806, 2752, 16928, 864, 3700, 6536, 1812, 868, 11264, 870, 13156, 8432, 1818, 29224, 1820, 876, 3712, 17888, 1824, 880, 22616, 882, 28288, 4664, 1830, 10336, 16952, 888, 3724, 2780, 1836, 4672, 11288, 894, 1840, 896, 1842, 44368, 5624, 900, 26416, 21692, 1848, 8464, 3740, 906, 5632, 27368, 1854, 910, 1856, 912, 15088, 14144, 1860, 27376, 3752, 918, 9424, 920, 1866, 6592, 13208, 924, 1870, 2816, 1872, 928, 7544, 930, 1876, 4712, 1878, 10384, 1880, 936, 7552, 2828, 1884, 940, 3776, 942, 1888, 27404
I arranged them by residue mod 945. This gives a direct proof that odd numbers > 72495 are the sum of two abundant numbers; you'd need a bit of finite checking to see that the bound can be improved to 20161. Computing the list took 40ms.
Of course I'm not sure that we need this proof, but if we do we probably don't need to include the list explicitly. - Charles R Greathouse IV 02:49, 23 September 2016 (UTC)
- My two cents: that might be worth a sequence entry in OEIS Main and a Theorem 2A here with a remark that "the proof is along the lines of the proof of Theorem 2, using 945 and the numbers of A280xxx." - Alonso del Arte 21:40, 23 September 2016 (UTC)
- I think A048242 is sufficient -- this finite sequence above is just one of many ways to prove that assertion, and not even a particularly clever one -- but I do like your wording suggestion, wherever we do put it. - Charles R Greathouse IV 07:15, 24 September 2016 (UTC)
Re: Density of abundant numbers
How did you get
in:
- at least (since 6 and 28 are perfect).
So that I could see why
is wrong. —
Daniel Forgues 23:44, 25 September 2016 (UTC)
- Oops, you're right: (I should not have done this mistake.)
- at least (since 6 and 28 are perfect).
- — Daniel Forgues 02:48, 30 September 2016 (UTC)
- Somehow that triggers me to write 5/30, 40/210, 506/2310, ... but there is no 5,40,506 in OEIS. Unrelated, the page is in good shape at the moment, maybe it could be promoted to "possible link target from OEIS". –Frank Ellermann 03:32, 20 October 2017 (UTC)