A063769 (and others)

Fri Aug 4 23:07:18 CEST 2006

It is widely, though not universally, conjectured that beyond a certain 
(large) point most numbers have unbounded aliquot sequences.  This, of 
course, is directly counter to the older conjecture that none of them 
are unbounded.  So the fact that many sequences appear unbounded does 
not by itself mean that you aren't going far enough.

Still, there are aliquot sequences that reach a large number of digits 
before returning to earth; 10^7 is certainly not bit enough.

-----
Looking at this, I noticed that the comment in A080907 implies that no 
number can have an aliquot sequence that terminates in a cycle of more 
than one element unless it is part of that cycle.  I suspect the author 
did not intend that implication; I would be very surprised if it was 
true.

I also noticed in A115350 that "Catalan's conjecture establishes that 
every aliquot sequence terminates in a prime number followed by 1, a 
perfect number, a friendly pair or an aliquot cycle."  Especially 
considering that it is quite likely that Catalan's conjecture is false, 
it is strange to talk about it "establishing" anything.  It would be 
better to use the word "is" rather than "establishes" here.  There are 
other problems here, too - claiming 43 is over 7% of all values can 
only be based on a small sample (there are only small samples of the 
integers), since no such theorem has been proven; the size of the 
sample should be noted.  And concerning "untouchable" primes "like 2 
and 5" - these are almost certainly the only untouchable primes.

Franklin T. Adams-Watters

-----Original Message-----
From: Joshua Zucker <joshua.zucker at gmail.com>

I am trying to extend A063769 -- numbers whose aliquot sequences end 
in perfect numbers -- and I wonder a few things. 

1) there are a lot of numbers whose aliquot sequences get pretty big. 
How big should they have to get before I go along with the comment in 
the existing sequence about 276, that it "seems unlikely" that they'll 
ever reach a perfect number? I arbitrarily took 10^7 as my bound, but 
that left about 10% of numbers (1074 numbers under 10,000) hitting the 
bound. So I tried 10^8 and found 1065 --- increasing that bound by a 
factor of 10 didn't help much. I suppose I could use a more efficient 
factorization algorithm [right now I just check every number up to the 
square root] and try bigger bounds, but it sure looks like a lot of 
these are headed to the stratosphere... 

2) is there a reason perfect numbers themselves shouldn't be included? 

3) what about 0 and 1? 

Thanks, 
--Joshua Zucker 

PS: here's what I get for subsequent terms, depending on the answers 
to the above questions -- so this is a candidate for a possible 
extension to A063769 

0 1 6 25 28 95 119 143 417 445 496 565 608 650 652 675 685 783 790 909 
913 1177 1235 1294 1441 1443 1574 1595 1633 1715 1717 1778 2162 2173 
2195 2225 2387 2541 2581 2582 2725 2863 3142 3277 3311 3337 3528 3575 
3693 3888 3899 3999 4141 4317 4535 4717 4739 4763 4775 4897 5158 5243 
5603 5605 5855 6099 6123 6145 6663 6672 7025 7149 7227 7396 7403 7527 
7543 7587 7856 7885 7923 8128 8135 8470 8477 8545 8593 8785 9035 9299 
9653 9683 9699 9921 9997 

and here's a sequence I was suprised not to find in OEIS -- maybe an 
indication that I messed something up? -- of terms whose aliquot 
sequence eventually reaches a cycle. Submitted just now as A121507, 
and the same sequence omitting the numbers which are themselves part 
of a cycle as A121508. 
220 284 562 1064 1184 1188 1210 1308 1336 1380 1420 1490 1604 1690 
1692 1772 1816 1898 2008 2122 2152 2172 2362 2542 2620 2630 2652 2676 
2678 2856 2924 2930 2950 2974 3124 3162 3202 3278 3286 3332 3350 3360 
3596 3712 3750 3850 3938 3944 4118 4156 4180 4372 4404 4448 4522 4790 
4840 4930 5020 5024 5078 5150 5234 5380 5564 5622 5630 5634 5784 5854 
5890 5900 5916 5942 5960 6028 6036 6232 6242 6280 6368 6420 6532 6612 
6658 6694 6808 6872 6876 7006 7106 7120 7130 7180 7186 7294 7380 7418 
7514 7520 7524 7540 7694 7796 7860 7930 7940 8076 8104 8306 8320 8354 
8480 8736 8776 8784 8924 9038 9052 9068 9204 9322 9332 9370 9464 9484 
9550 9572 9574 9620 

Most of those eventually come down to one of the amicable pairs, 
A063990 
220 284 
1184 1210 
2620 2924 
5020 5564 
6232 6368 
but some of them come down to other cycles. 
A072891: 12496 14288 15472 14536 14264 

and, cool! I rediscovered A072890: 
14316 19116 31704 47616 83328 177792 295488 629072 589786 294896 
358336 418904 366556 274924 275444 243760 376736 381028 285778 152990 
122410 97946 48976 45946 22976 22744 19916 17716 

and here's the list of the thousand or so numbers whose aliquot 
sequences get "pretty big" (greater than 10^8) 

138 150 168 222 234 276 306 312 396 528 552 564 570 660 696 702 720 
726 780 828 840 858 870 888 936 960 966 978 990 996 1044 1062 1074 
1086 1098 1104 1134 1146 1158 1170 1218 1230 1248 1266 1278 1290 1302 
1314 1316 1320 1326 1338 1350 1356 1372 1392 1398 1410 1422 1428 1440 
1464 1476 1482 1488 1506 1512 1518 1521 1530 1542 1554 1560 1572 1578 
1590 1596 1620 1632 1650 1662 1674 1686 1698 1710 1722 1734 1758 1770 
1794 1806 1818 1820 1836 1842 1848 1854 1860 1866 1872 1876 1878 1890 
1896 1902 1914 1920 1932 1938 1944 1950 1962 1986 1992 1998 2010 2040 
2046 2058 2082 2088 2094 2106 2124 2136 2142 2154 2160 2166 2190 2202 
2208 2214 2226 2232 2238 2250 2256 2280 2292 2298 2310 2322 2328 2334 
2340 2346 2352 2356 2360 2370 2382 2394 2406 2418 2424 2430 2442 2448 
2478 2480 2484 2502 2514 2526 2538 2544 2550 2562 2568 2580 2604 2622 
2640 2658 2664 2670 2682 2712 2716 2718 2724 2742 2754 2760 2772 2784 
2802 2814 2826 2828 2838 2850 2864 2874 2880 2884 2886 2898 2904 2910 
2912 2922 2934 2940 2946 2952 2958 2970 2976 2982 3000 3018 3025 3030 
3036 3040 3048 3050 3060 3084 3102 3120 3126 3138 3144 3150 3156 3168 
3172 3174 3192 3198 3204 3210 3222 3234 3246 3258 3264 3270 3282 3288 
3294 3306 3312 3318 3336 3366 3378 3390 3408 3430 3432 3444 3450 3456 
3462 3472 3474 3480 3486 3492 3498 3510 3516 3522 3534 3552 3556 3564 
3582 3588 3612 3630 3636 3660 3678 3690 3696 3702 3714 3720 3726 3738 
3746 3762 3770 3774 3780 3786 3790 3796 3798 3810 3822 3828 3834 3836 
3840 3846 3858 3864 3870 3876 3882 3892 3894 3906 3912 3918 3930 3936 
3942 3948 3960 3990 4002 4008 4016 4026 4068 4086 4092 4098 4107 4110 
4116 4122 4128 4134 4140 4144 4146 4152 4158 4170 4200 4206 4218 4224 
4230 4236 4278 4290 4296 4302 4314 4320 4326 4350 4362 4374 4380 4386 
4392 4396 4422 4434 4440 4446 4452 4458 4464 4470 4482 4488 4494 4500 
4506 4512 4518 4520 4530 4560 4564 4566 4578 4590 4602 4614 4620 4626 
4632 4638 4640 4644 4648 4650 4668 4674 4686 4704 4710 4714 4716 4736 
4746 4758 4770 4776 4782 4788 4794 4800 4806 4812 4818 4830 4842 4848 
4854 4860 4866 4878 4902 4914 4920 4926 4938 4950 4954 4956 4962 4974 
4986 4992 4998 5010 5028 5034 5044 5046 5058 5064 5076 5082 5088 5094 
5106 5118 5130 5148 5160 5178 5180 5190 5208 5214 5220 5226 5232 5238 
5250 5280 5286 5292 5298 5304 5310 5352 5364 5370 5394 5400 5406 5418 
5426 5430 5432 5436 5442 5448 5454 5456 5466 5478 5480 5490 5502 5508 
5514 5520 5526 5536 5538 5548 5574 5586 5598 5610 5640 5646 5658 5664 
5670 5676 5680 5682 5688 5694 5718 5722 5730 5736 5740 5742 5748 5754 
5760 5766 5778 5796 5808 5818 5820 5826 5838 5840 5852 5856 5862 5870 
5872 5874 5880 5886 5898 5904 5908 5910 5922 5928 5934 5940 5946 5958 
5964 5970 5982 5994 6024 6058 6072 6074 6076 6078 6090 6102 6108 6114 
6126 6132 6138 6150 6160 6170 6174 6184 6186 6188 6198 6204 6210 6222 
6228 6234 6240 6244 6246 6252 6258 6264 6270 6276 6282 6288 6294 6300 
6304 6306 6318 6328 6330 6338 6342 6348 6360 6366 6372 6378 6390 6396 
6402 6408 6414 6426 6438 6444 6448 6450 6462 6474 6480 6486 6504 6522 
6534 6546 6552 6558 6570 6600 6604 6618 6624 6630 6632 6636 6648 6654 
6656 6660 6666 6680 6688 6690 6692 6700 6720 6726 6732 6738 6744 6748 
6750 6752 6756 6768 6774 6786 6792 6804 6810 6822 6832 6834 6846 6882 
6894 6902 6938 6940 6948 6960 6966 6972 6978 6984 6990 7002 7008 7014 
7020 7026 7028 7032 7038 7044 7062 7074 7080 7084 7086 7092 7098 7110 
7122 7134 7158 7170 7172 7182 7192 7206 7208 7216 7218 7224 7225 7230 
7242 7254 7266 7296 7302 7308 7314 7320 7326 7338 7344 7350 7352 7360 
7362 7364 7368 7372 7388 7392 7412 7416 7420 7422 7434 7440 7446 7458 
7464 7482 7494 7504 7506 7518 7530 7532 7534 7542 7554 7560 7566 7570 
7578 7584 7586 7588 7590 7596 7602 7614 7632 7638 7644 7656 7662 7664 
7666 7674 7676 7680 7684 7686 7690 7692 7698 7710 7712 7752 7756 7770 
7778 7800 7806 7812 7818 7828 7830 7842 7852 7854 7866 7868 7878 7890 
7902 7908 7920 7922 7924 7950 7956 7962 7968 7974 7980 7982 7986 7992 
7998 8015 8016 8020 8022 8024 8026 8028 8034 8040 8046 8052 8056 8058 
8064 8068 8070 8080 8082 8084 8088 8092 8094 8112 8118 8120 8130 8136 
8142 8144 8148 8154 8160 8164 8172 8176 8184 8190 8204 8208 8232 8238 
8250 8260 8262 8274 8280 8286 8288 8298 8300 8310 8316 8322 8328 8334 
8340 8346 8352 8358 8364 8370 8372 8388 8396 8408 8412 8418 8424 8427 
8430 8432 8440 8442 8460 8472 8476 8484 8490 8496 8512 8514 8520 8526 
8532 8538 8540 8544 8550 8568 8580 8588 8596 8598 8610 8622 8628 8632 
8648 8650 8652 8658 8664 8682 8694 8706 8718 8720 8730 8744 8748 8754 
8760 8766 8802 8814 8826 8838 8850 8862 8864 8872 8876 8880 8886 8898 
8904 8910 8922 8932 8934 8944 8946 8952 8958 8964 8970 8976 8988 8994 
8996 9000 9006 9008 9018 9024 9028 9030 9034 9036 9044 9048 9054 9096 
9100 9108 9114 9120 9122 9126 9136 9138 9150 9156 9168 9180 9186 9192 
9196 9198 9200 9210 9222 9234 9240 9244 9246 9252 9258 9270 9274 9280 
9282 9288 9294 9306 9318 9320 9330 9336 9342 9352 9354 9360 9366 9372 
9378 9380 9390 9402 9412 9414 9420 9424 9426 9436 9438 9450 9456 9462 
9466 9468 9474 9480 9486 9492 9534 9540 9558 9564 9570 9576 9582 9588 
9592 9594 9606 9614 9618 9660 9680 9684 9696 9702 9708 9712 9714 9716 
9720 9726 9728 9738 9750 9762 9768 9772 9774 9786 9798 9810 9816 9822 
9828 9834 9840 9844 9848 9852 9858 9870 9888 9894 9902 9906 9912 9916 
9918 9924 9928 9930 9936 9942 9950 9954 9958 9980 